Blowups and Resolution Apr 2013

Blowups and Resolution

Herwig Hauser

To the memory of Sheeram Abhyankar,with great respect.

The present manuscript originates from a series of lectures the author1 gave at theClay Summer School on Resolution of Singularities at Obergurgl in the TyroleanAlps in June 2012. A hundred young students gathered for four weeks to hearand learn about resolution of singularities. Their interest and dedication becameessential for the success of the school.

The idealistic reader for this article is an algebraist or geometer having a rudi-mentary acquaintance with the main techniques and results in resolution of singu-larities – wishing to get quick and concrete reference about specific topics in thefield. As such, the article is modelled like a dictionary and not particularly suitedto be read from the beginning to the end (except for those who like to read dictio-naries). Also, in order to enable to read and understand selected portions of thetext without having to browe the whole earlier material, a certain redundance ofdefinitions and assertions had to be accepted.

Background informations on the historic development and the motivations be-hind the various constructions can be found in the cited literature, especially in[Obe00], [Hau03], [Hau10a], [FH10], [Cut04], [Kol07], [Lip75]. Complete proofs ofseveral more technical results can be found in [EH02].

All statements are formulated for varieties and morphism between them. Incertain cases, the respective statements for schemes are indicated separately.

Each chapter concludes with a broad selection of examples, ranging from com-putational exercises to suggestions for additional material which could not be cov-ered in the text. Some more challenging problems are marked accordingly. Theexamples should be especially useful for people planning to give a graduate courseon the resolution of singularities. Occasionally the examples repeat or specializestatements which have appeared in the text and which are worth to be done per-sonally before looking at the given proof.

Several results appear without proof, due to lack of time and energy of theauthor. Precise references are given whenever possible. The various survey articlescontain complementary bibliography.

We are indebted to the Clay Mathematics Institute for choosing resolution ofsingularities as the topic of the 2012 summer school. It has been a particular plea-sure to cooperate in this endeavour with its research director David Ellwood, whose

1Supported in part by the Austrian Science Fund FWF within the project P-21461.1

2 HERWIG HAUSER

enthusiasm and interpretation of the school largely coincided with the approach ofthe organizers, thus creating a wonderful working atmosphere. His sensitiveness ofhow to plan and realize the event has been exceptional.

The CMI director Jim Carlson and the CMI secretary Julie Feskoe supportedvery efficiently the preparation and realization of the school.

Xudong Zheng provided a preliminary write-up of the lectures, Stefan Perlegaand Eleonore Faber completed several missing parts in earlier drafts of the man-uscript. Perlega worked out many technical details, and Faber produced the twovisualizations. Barbara Beeton from the AMS took care of the TeX-layout. All thiswas very helpful, thanks!

1. Lecture I: An Incomprehensible Example

Let X be the zeroset in affine three space A3 of the polynomial

f = 27x2y3z2 + (x2 + y3 − z2)3

over a field K of characteristic different from 2 and 3. This is an algebraic surfacewith possibly singular points and curves, and certain symmetries. For instance, theorigin 0 is singular on X, and X is symmetric with respect to the automorphismsof A3 given by replacing x by −x or z by −z, and also by replacing y by −y whileinterchanging x with z. Sending x, y and z to t3x , t2y and t3z for t ∈ K alsopreserves X. See figure 1 for a plot of the real points of X. The intersections ofX with the three coordinate hyperplanes of A3 are given as the zerosets of theequations

x = (y3 − z2)3 = 0,y = (x2 − z2)3 = 0,z = (x2 + y3)3 = 0.

These intersections are plane curves: two perpendicular cusps lying in the xy- andyz-plane, respectively the union of the two diagonals in the xz-plane. The singularlocus Sing(X) of X is given as the zeroset of the partial derivatives of f . This yieldsfor Sing(X) the equations

x · [9y3z2 + (x2 + y3 − z2)2] = 0,

y2 · [9x2z2 + (x2 + y3 − z2)2] = 0,z · [9x2y3 − (x2 + y3 − z2)2] = 0.

Therefore the singular locus of X has six irreducible components, defined by

x = y3 − z2 = 0,

z = x2 + y3 = 0,y = x+ z = 0,y = x− z = 0,

x2 − y3 = x+√−1z = 0,

x2 − y3 = x−√−1z = 0.

The first four components of Sing(X) coincide with the four curves given by thethree coordinate hyperplane sections of X. The last two components are planecusps in the hyperplanes given by x±

√−1z = 0. At points a 6= 0 on the first two

singular components of Sing(X), the intersections of X with a plane through a andtransversal to the component are again cuspidal curves.

BLOWUPS AND RESOLUTION 3

Figure 1. The surface Camelia: 27x2y3z2 + (x2 + y3 − z2)3 = 0.

Consider now the surface Y in A4 which is given as the cartesian product C×Cof the plane cusp C : x2 − y3 = 0 in A2 with itself. It is defined by the equationsx2 − y3 = z2 − w3 = 0. The singular locus Sing(Y ) is the union of the two cuspsC1 = C × 0 and C2 = 0 × C defined by x2 − y3 = z = w = 0, respectivelyx = y = z2 − w3 = 0. The surface Y admits the parametrization

γ : A2 → A4, (s, t) 7→ (s3, s2, t3, t2).

The image of γ is Y . The composition of γ with the linear projection

π : A4 → A3, (x, y, z, w) 7→ (x,−y + w, z)

yields the map

δ = π ◦ γ : A2 → A3, (s, t) 7→ (s3,−s2 + t2, t3).

Replacing in f the variables x, y, z by s3, −s2 + t2, t3 gives 0. This shows that theimage of δ lies in X. As X is irreducible of dimension 2 and δ has rank 2 outside0 the image of δ is dense in X. Therefore the image of Y under π is dense in X:This interprets X as a contraction of Y by means of the projection π from A4 toA3. The two surfaces are not isomorphic; for instance, their singular loci have adifferent number of components. The simple geometry of Y as a cartesian productof two plane curves is scrambled up under the projection.

The point blowup of Y in the origin produces a surface Y1 whose singular locushas two components. They map to the two components C1 and C2 of Sing(Y )and are regular and transversal to each other. The blowup X1 of X at 0 will stillbe the image of Y1 under a suitable projection. The four singular components ofSing(X) will become regular in X1 and will either meet pairwise transversally ornot at all. The two regular components of Sing(X) will remain regular in X1 butwill no longer meet each other, cf. figure 2.

The point blowup of Y1 in the intersection point of the two curves of Sing(Y1)separates the two curves and yields a surface Y2 whose singular locus consists oftwo disjoint regular curves. Blowing up these separately yields a regular surface Y3

and thus resolves the singularities of Y . The resolution of the singularities of X ismore complicated, see the examples below.

4 HERWIG HAUSER

Figure 2. The surface obtained from Camelia by a point blowup .

Example 1.1. Show that the surface X defined in A3 by 27x2y3z2 +(x2 +y3−z2)3 = 0 is the image of the cartesian product Y of the cusp C : x3 − y2 = 0 withitself under the projection from A4 to A3 given by (x, y, z, w) 7→ (x,−y + w, z).

Example 1.2. Find additional symmetries of X aside from those mentionedin the text.

Example 1.3. Produce a real visualization of the surface obtained from Cameliaby replacing in the equation z by

√−1z. Determine the components of the singular

locus.

Example 1.4. Consider at the point a = (0, 1, 1) of X the plane P : 2y+3z = 5through a. It is transversal to the component of the singular locus of X passingthrough a (i.e., this component is regular at a and P intersects its tangent line ina point). Determine the singularity of X ∩ P at a. The normal vector (0, 2, 3) toP is the tangent vector at t = 1 of the parametrization (0, t2, t3) of the componentof Sing(X) defined by x = y3 − z2.

Example 1.5. The point a on X with coordinates (1, 1,√−1) is a singular

point of X, and a non-singular point of the curve of Sing(X) passing through it. Aplane transversal to Sing(X) at a is given e.g. by

P : 3x+ 2y + 3√−1 · z = 9.

Determine the singularity of X ∩ P at a. The normal vector (3, 2, 3√−1) to P is

the tangent vector at s = 1 of the parametrization (s3, s2,√−1s3) of the curve of

Sing(X) through a defined by x2 − y3 = x−√−1z = 0.

Example 1.6. Blow up X and Y in the origin, describe exactly the geometryof the transforms X1 and Y1 and produce visualizations of X and X1 over R in allcoordinate charts. For Y1 the equations are in the x-chart 1− xy3 = z3x−w2 = 0and in the y-chart x2 − y = z3y − w2 = 0.

Example 1.7. Blow up X1 and Y1 along their singular loci. Compute a reso-lution of X1 and compare it with the resolution of Y1.


2. Lecture II: Varieties and Schemes

The following summary of basic concepts shall clarify the terminology used in latersections. Detailed definitions are available in [Mum99, Sha94, Har77, EH00, Liu02,Kem11, GW10, Gro61, ZS75, Nag75, Mat89, AM69].

Definition 2.1. Write An = AnK for the affine n-space over some field K,equipped with the Zariski topology whose closed sets are defined as the algebraicsubsets of An, i.e., as the zerosets V (I) of ideals I of K[x1, . . . , xn]. The coordinatering K[An] of An is the polynomial ring K[x1, . . . , xn] in n variables over K. Theset of prime ideals of K[x1, . . . , xn] is denoted by Spec(K[x1, . . . , xn]) and equippedwith the Zariski topology whose closed sets V (I) are formed by the prime idealscontaining a given ideal I of K[x1, . . . , xn]. This gives rise to the scheme-theoreticversion of affine space An.

Definition 2.2. A polynomial map g : An → Am is given by a vector g =(g1, . . . , gm) of polynomials gi(x) ∈ K[x1, . . . , xn]. It induces a K-algebra homo-morphism f∗ : K[y1, . . . , ym] → K[x1, . . . , xn] sending h to h ◦ f . A rationalmap g : An → Am is given by a vector g = (g1, . . . , gm) of rational functionsgi(x) ∈ K(x1, . . . , xn) = Quot(K[x1, . . . , xn]). A rational map need not be definedon whole An; it is only defined on the complement of the union of the zerosets ofthe denominators of the functions gi.

Definition 2.3. An affine algebraic variety X is a subset of affine space Andefined as the zeroset V (I) of a radical ideal I of K[x1, . . . , xn] and equipped withthe topology induced by the Zariski topology of An. The ideal I need not beprime, hence X is not required to be irreducible. The affine coordinate ring of Xis the factor ring K[X] = K[x1, . . . , xn]/I. As I is radical, K[X] is reduced, i.e.,has no nilotent elements. If X is irreducible, I is a prime ideal and K[X] is anintegral domain. The function field of an irreducible variety X is the quotient fieldK(X) = Quot(K[X]). The local ring of a variety X at a point a is the localizationOX,a = K[X]mX,a

of K[X] at the maximal ideal mX,a defining a in X.

Definition 2.4. A quasi-affine variety is an open subset of an affine variety.A principal open subset of X is the complement in X of the zeroset V (g) of a singlepolynomial g.

Definition 2.5. Write Pn = PnK for the projective n-space over some field K,equipped with the Zariski topology whose closed sets are defined as the algebraicsubsets of Pn, i.e., as the zerosets V (I) of homogeneous ideals I of K[x0, . . . , xn].The projective space is covered by the affine open subsets Ui ' An formed bythe points whose i-th projective coordinate does not vanish. The homogeneouscoordinate ring of Pn is the graded polynomial ring K[x0, . . . , xn] in n+ 1 variablesover K, the grading given by the degree. The set of homogeneous prime ideals ofK[x0, . . . , xn] not containing the ideal (x0, . . . , xn) is denoted by Proj(K[x0, . . . , xn])and equipped with the Zariski topology whose closed sets V (I) are formed by thehomogeneous prime ideals containing a given ideal I of K[x0, . . . , xn]. This givesrise to the scheme-theoretic version of projective space Pn.

Definition 2.6. A projective algebraic variety X is a subset of projectivespace Pn defined as the zeroset V (I) of a homogeneous radical ideal I = IX ofK[x1, . . . , xn] and equipped with the Zariski topology. The projective coordinate

6 HERWIG HAUSER

ring of X is the factor ring K[X] = K[x1, . . . , xn]/I equipped with the gradinggiven by degree.

Definition 2.7. A quasi-projective variety is an open subset of a projectivevariety. A principal open subset is the complement of the zeroset V (g) of a singlehomogeneous polynomial g.

Remark 2.8. Abstract algebraic varieties are obtained by gluing affine alge-braic varieties along principal open subsets, cf. [Mum99] I, §3, §4, [Sha94] V, §3.This allows to develop the category of algebraic varieties with the usual construc-tions therein.

Definition 2.9. Let X and Y be two affine algebraic varieties X ⊂ Am andY ⊂ An, and let U be an open subset of X. A regular map from U to Y isa map f : U → An sending U into Y with components rational functions whosedenominators do not vanish on U . A morphism f : X → Y is a regular map definedon whole X. It induces a K-algebra homomorphism f∗ : K[Y ] → K[X] betweenthe coordinate rings, which, in turn, determines f . Over algebraically closed fields,a morphism is the restriction to X of a polynomial map f : Am → An sending Xinto Y , i.e., such that f∗(IY ) ⊂ IX [Har77].

Definition 2.10. A rational map f : X → Y between affine varieties is amorphism f : U → Y defined on some dense open subset U of X. Equivalently,a rational map is given by a K-algebra homomorphism αf : K(Y ) → K(X). Abirational map f : X → Y is a regular map f : U → Y on some dense open subsetU of X such that V = f(U) is open in Y and such that fU : U → V is a regularisomorphism, i.e., admits an inverse morphism. In this case U and V are calledbiregularly isomorphic, and X and Y are birationally isomorphic. Equivalently, abirational map is given by an isomorphism αf : K(Y ) → K(X) of the functionfields. A birational morphism f : X → Y is a birational map which is defined onwhole X, i.e., a regular morphism f : X → Y with rational inverse defined on adense open subset of Y .

Definition 2.11. A morphism f : X → Y between algebraic varieties is calledseparated if the diagonal ∆ ⊂ X ×Y X is closed in the fibre product. A morphismf : X → Y between varieties is proper if it is separated and universally closed, i.e.,for any variety Z and morphism Z → Y the induced morphism g : X ×Y Z → Y isclosed.

Definition 2.12. The germ of a variety X at a point a, denoted by (X, a),is the equivalence class of on open neighbourhoods U of a in X where two neigh-bourhoods of a are said to be equivalent if they coincide on a (possibly smaller)neighbourhood of a. Equivalently, the germ of a variety is given by the localizationOX,a = K[X]mX,a

of the coordinate ring K[X] of X at the maximal ideal mX,a ofK[X] defining a in X.

Definition 2.13. Let X and Y be two varieties, and let a and b be points ofX and Y . The germ of a morphism f : (X, a)→ (Y, b) at a is the equivalence classof a morphism f : U → Y defined on an open neighbourhood U of a in X andsending a to b; here, two morphisms defined on neighbourhoods of a in X are saidto be equivalent if they coincide on a (possibly smaller) neighbourhood of a. Themorphism f : U → Y is called a representative of f on U . Equivalently, the germ ofa morphism is given by a local K-algebra homomorphism αf = f∗ : OY,b → OX,a.


Definition 2.14. The tangent space TaX to a variety X at a point a is definedas the K-vector space

TaX = (mX,a/m2X,a)∗ = Hom(mX,a/m2

X,a,K)

with mX,a ⊂ OX,a the maximal ideal of a. The tangent map Taf : TaX → TbYof a morphism f : X → Y sending a point a ∈ X to b ∈ Y is defined as the mapinduced naturally by f∗ : OY,b → OX,a.

Definition 2.15. The embedding dimension embdimaX of a variety X at apoint a is defined as the K-dimension of TaX.

Definition 2.16. An affine scheme X is a commutative ring R with 1, calledthe coordinate ring of X or ring of global sections. The set Spec(R), also denotedby X and called the spectrum of R, is defined as the set of prime ideals of R. Here,R does not count as a prime ideal, but 0 does if it is prime, i.e., if R is an integraldomain. The spectrum is equipped with a topology, the Zariski topology, whoseclosed sets V (I) are formed by the prime ideals containing a given ideal I of R, andwith a sheaf of rings OX , the structure sheaf of X, whose stalks are the localizationsof R at prime ideals [Mum99, Har77, Sha94]. An affine scheme of finite type oversome field K is an affine scheme whose coordinate ring R is a finitely generatedK-algebra, i.e., a factor ring K[x1, . . . , xn]/I of a polynomial ring by some arbitraryideal I.

Definition 2.17. Let X be an affine scheme of coordinate ring R. A closedsubscheme Y of X is a factor ring S = R/I of R by some ideal I of R, togetherwith the canonical homomorphism R → R/I. The set Y = Spec(R/I) of primeideals of R/I can be identified with the closed subset V (I) of X = Spec(R) ofprime ideals of R containing I. It carries the topology induced by the Zariskitopology of X. A point a of X is a prime ideal I of R, considered as an elementof Spec(R). To a point one associates a closed subscheme of X via the factor ringR/I. The point a is closed if I is a maximal ideal of R. A principal open subset ofX is the affine scheme U defined by the ring of fractions Rg of R with respect tothe multiplicatively closed set {1, g, g2, . . .} for some non-zero divisor g ∈ R. Thenatural ring homomorphism R→ Rg sending f to f/1 interprets U = Spec(Rg) asan open subset of X = Spec(R).

Remark 2.18. Abstract schemes are obtained by gluing affine schemes alongprincipal open subsets [Mum99] II, §1, §2, [Har77] II.2, [Sha94] V, §3. For localconsiderations, one can often restrict to affine schemes.

Definition 2.19. A morphism f : X → Y between affine schemes X =Spec(R) and Y = Spec(S) is a ring homomorphism α = αf : S → R. It in-duces a continuous map Spec(R) → Spec(S) by sending a prime ideal I of R tothe prime ideal α−1(I) of S. Morphisms between arbitary schemes are defined bychoosing coverings by affine schemes and defining the morphism locally. A rationalmap f : X → Y between schemes is a morphism f : U → Y defined on a dense opensubscheme U of X. It need not be defined on whole X. A rational map is birationalif the morphism f : U → Y admits on a dense open subscheme V of Y an inversemorphism V → U , i.e., if f|U : U → V is an isomorphism. A birational morphismf : X → Y is a morphism f : X → Y which is also a birational map, i.e., inducesan isomorphism U → V of dense open subschemes. In contrast to birational maps,

8 HERWIG HAUSER

a birational morphism is defined on whole X, while its inverse map is only definedon a dense open subset of Y .

Definition 2.20. Let R =⊕∞

i=0Ri be a graded ring. The set X = Proj(R) ofhomogeneous prime ideals of R not containing the irrelevant ideal M = ⊕∞i≥1Ri isequipped with a topology, the Zariski topology, and with a sheaf of rings OX , thestructure sheaf of X [Mum99, Har77, Sha94]. It thus becomes a scheme. Typically,R is generated as an R0-algebra by the homogeneous elements g ∈ R1 of degree 1.An open covering of X is then given by the affine schemes Xg = Spec(R◦g), whereR◦g denotes, for any non-zero g ∈ R1, the subring of elements of degree 0 in the ringof fractions Rg.

Remark 2.21. A graded ring homomorphism S → R induces a morphism ofschemes Proj(R)→ Proj(S).

Definition 2.22. Equip the polynomial ring K[x0, . . . , xn] with the naturalgrading given by the degree. The scheme Proj(K[x0, . . . , xn]) is called n-dimensio-nal projective space over K, denoted by Pn = PnK. As a variety, it coincides withthe classical definition of projective space as the quotient Pn = (An+1 \ {0})/K∗,as introduced in def. 2.5.

Definition 2.23. A morphism f : X → Y is called projective if it factors, forsome k, into a closed embedding X ↪→ Y ×Pk followed by the projection Y ×Pk → Yon the first factor [Har77] II.4, p. 103.

Definition 2.24. Let R be a ring and I an ideal of R. The powers Ik of Iinduce natural homomorphisms R/Ik+1 → R/Ik. The I-adic completion of R isthe inverse limit R = lim←−R/I

k, together with the canonical homomorphism R→ R.If R is a local ring with maximal ideal m, the m-adic completion R is called thecompletion of R. For an R-module M , one defines the I-adic completion M of Mas the inverse limit M = lim←−M/Ik ·M .

Lemma 2.25. Let R be a noetherian ring with prime ideal I and I-adic com-pletion R.

(1) If J is another ideal of R with I ∩ J-adic completion J , then J = J · Rand R/J ' R/J .

(2) If J1, J2 are two ideals of R and J = J1 · J2, then J = J1 · J2.(3) If R is a local ring with maximal ideal m, the completion R is a noetherian

local ring with maximal ideal m = m ·R, and m ∩R = m.(4) If J is an arbitrary ideal of a local ring R, then J ∩R =

⋂i≥0(J + mi).

(5) Passing to the I-adic completion defines an exact functor on finitely gen-erated R-modules.

(6) R is a flat R-algebra.(7) If M a finitely generated R-module, then M = M ⊗R R.

Proof. (1) By [ZS75] VIII, Thm. 6, Cor. 2, p. 258, it suffices to verify that J isclosed in the I∩J-adic topology. The closure J of J equals J =

⋂i≥0(J+(I∩J)i) =

J by [ZS75] VIII, Lem. 1, p. 253.(2) By definition, J = J1 · J2 · R = J1 · R · J2 · R = J1 · J2.


(3), (4) The ring R is noetherian and local by [AM69] Prop. 10.26, p. 113, andProp. 10.16, p. 109. The ideal I∩R equals the closure I of I in the m-adic topology.Thus, I ∩R =

⋂i≥0(I + mi) and m ∩R = m.

(5) – (7) [AM69] Prop. 10.12, p. 108, Prop. 10.14, p. 109, Prop. 10.14, p. 109. �

Definition 2.26. Let X be a variety or a scheme and let a be a point ofX. The local ring OX,a of X at a is equipped with the mX,a-adic topology whosebasis of neighbourhoods of 0 is given by the powers mk

X,a of mX,a. The inducedcompletion OX,a of OX,a is called the complete local ring of X at a [Nag75] II,[ZS75] VIII, §1, §2. The scheme defined by OX,a is called the formal neighbourhoodor formal germ of X at a, denoted by (X, a). A formal subvariety (Y , a) of (X, a)is a factor ring OX,a/I by an ideal I of OX,a. A map f : (X, a) → (Y , b) betweentwo formal germs is defined as a local algebra homomorphism αf : OY,b → OX,a.It is also called a formal map between X and Y at a. A formal coordinate changeof X at a is an automorphism of OX,a.

Remark 2.27. The inverse function theorem does not hold for algebraic vari-eties and regular maps between them, but it holds in the category of formal germsand formal maps: A formal map f : (X, a)→ (Y , b) is an isomorphism if and onlyif its tangent map Taf : TaX → TbY is a linear isomorphism.

Definition 2.28. A morphism f : X → Y between varieties is called etaleif for all a ∈ X the induced maps of formal germs f : (X, a) → (Y , f(a)) areisomorphisms, or, equivalently, if all tangent maps Taf of f are isomorphisms.

Definition 2.29. A morphism f : X → Y between varieties is called smoothif for all a ∈ X the induced maps of formal germs f : (X, a) → (Y , f(a)) aresubmersions, i.e., the tangent maps Taf of f are surjective.

Remark 2.30. The category of formal germs and formal maps admits the usualconcepts and constructions as e.g. the decomposition of a variety in irreduciblecomponents, intersections of germs, inverse images, or fibre products. Similarly,when working of R, C or any complete valued field K, one can develop, based onrings of convergent power series, the category of analytic varieties and analyticspaces, respectively of their germs, and analytic maps between them [dJP00].

Remark 2.31. The natural ring homomorphism OX,a → OX,a defines a map(X, a)→ (X, a) of the formal neighbourhood into the germ of X at a when consid-ered as schemes.

Example 2.32. Compare the algebraic varieties X satisfying (X, a) ∼= (Ad, 0)for all a ∈ X (where ∼= stands for biregularly isomorphic Zariski-germs) with thosewhere the isomorphism is just formal. Varieties with the first property are calledplain, cf. def. 3.11.

Example 2.33. It can be shown that any complex regular and rational surface(i.e., regular and birationally isomorphic to affine space over C) is plain [BHSV08]Prop. 3.2. Show directly that X defined in A3 by x − (x2 + z2)y = 0 is plain byexhibiting a local isomorphism at 0 with A2. Is there an algorithm to constructsuch a local isomorphism for any complex regular and rational surface?

10 HERWIG HAUSER

Example 2.34. Let X be an algebraic variety and a ∈ X be a point. Whatis the difference between the concept of regular system of parameters in OX,a andOX,a?

Example 2.35. The formal neighbourhood of affine space An at a point a =(a1, . . . , an) is given by the formal power series ring OAn,a ' K[[x1−a1, . . . , xn−an]].If X ⊂ An is an affine algebraic variety defined by the ideal I of K[x1, . . . , xn],the formal neighbourhood (X, a) is given by the factor ring OX,a = OAn,a/I 'K[[x1−a1, . . . , xn−an]]/I, where I = I · OAn,a denotes the extension of I to OAn,a.

Example 2.36. The map f : A1 → A2 given by t 7→ (t2, t3) is a regularmorphism which induces a birational isomorphism onto the curve X in A2 definedby x3 = y2. The inverse (x, y) 7→ y/x is rational on X and regular on X \ {0}.

Example 2.37. The map f : A1 → A2 given by t 7→ (t2 − 1, t(t2 − 1)) is aregular morphism which induces a birational isomorphism onto the curve X in A2

defined by x2 + x3 = y2. The inverse (x, y) 7→ y/x is rational on X and regularon X \ {0}. The germ (X, 0) of X at 0 is not isomorphic to the germ (Y, 0) of theunion Y of the two diagonals in A2 defined by x2 = y2. The formal germs (X, 0)and (Y , 0) are isomorphic via the map (x, y) 7→ (x

√1 + x, y).

Example 2.38. The map f : A2 → A2 given by (x, y) 7→ (xy, y) is a birationalmorphism with inverse the rational map (x, y) 7→ (xy , y). The inverse defines aregular moprhism outside the x-axis.

Example 2.39. The maps ϕij : An+1 → An+1 given by

(x0, . . . , xn) 7→ (x0

xj, . . . ,

xi−1

xj, 1,

xi+1

xj, . . . ,

xnxj

).

are birational maps for each i, j = 0, . . . , n. They are the transition maps betweenthe affine charts Uj = Pn \ V (xj) ' An of projective space Pn.

Example 2.40. The maps ϕij : An → An given by

(x1, . . . , xn) 7→(x1

xj, . . . ,

xi−1

xj,

1xj,xi+1

xj, . . . ,

xj−1

xj, xixj ,

xj+1

xj, . . . ,

xnxj

)are birational maps for i, j = 1, . . . , n. They are the transition maps between theaffine charts of the blowup An ⊂ An × Pn−1 of An at the origin.

Example 2.41. The elliptic curve X defined in A2 by y2 = x3 − x is for-mally isomorphic at each point a of X to (A1, 0), whereas the germs (X, a) are notbiregularly isomorphic to (A1, 0).

Example 2.42. The projection (x, y) 7→ x from the hyperbola X defined in A2

by xy = 1 to the x-axis A1 is not proper, and the image of X is not closed.

Example 2.43. The curves X and Y defined in A2 by x3 = y2, respectivelyx5 = y2 are not formally isomorphic to each other at 0, whereas the curve Z definedin A2 by x3 + x5 = y2 is formally isomorphic to X at 0.


3. Lecture III: Singularities

Let X be an affine algebraic variety defined over a field K. Analogous concepts asthe ones given below can be defined for abstract varieties and schemes.

Definition 3.1. A noetherian local ring R with maximal ideal m is calleda regular ring if m can be generated by n elements, where n is the vector spacedimension of m/m2 over the residue field R/m, or, equivalently, the Krull dimensionof R.

Definition 3.2. Let R be a noetherian regular local ring with maximal idealm. A minimal set of generators of x1, . . . , xn of m is called a regular system ofparameters or local coordinates of R.

Remark 3.3. A regular system of parameters of a local ring R is also a regularsystem of parameters of its completion R, but not conversely, cf. ex. 3.27.

Definition 3.4. A point a of X is a regular or non-singular point of X if thelocal ring OX,a of X at a is a regular ring. Equivalently, the mX,a-adic completionOX,a of OX,a is isomorphic to the completion OAd,0 ' K[[x1, . . . , xd]] of the localring of some affine space Ad at 0. Otherwise a is called a singular point or a singu-larity of X. The set of singular points of X is denoted by Sing(X), its complementby Reg(X). The variety X is regular or non-singular if all its points are regular.Over perfect fields regular varieties are also called smooth.

Proposition 3.5. A subvariety X of a regular variety W is regular at a if andonly if there are local coordinates x1, . . . , xn of W at a so that X can be definedlocally at a by x1 = · · · = xk = 0 where k is the codimension of X in W at a.

Proof. [dJP00] Cor. 4.3.20, p. 155. �

Remark 3.6. Even though regular points are defined through the completionof the local rings, the regular parameter system as in the proposition can be takenin the local ring. The germ of a variety at a regular point need not be biregularlyisomorphic to the germ of an affine space at 0. This only holds for the formalneighbourhoods, cf. ex. 3.28.

Proposition 3.7. Assume that the field K is perfect. Let X be a hypersurfacedefined in a regular variety W by the square-free equation f = 0. The point a ∈ Xis singular if and only if all partial derivatives ∂x1f, . . . , ∂xnf of vanish at a.

Proof. [Zar47] Thm. 7, [Har77] I.5, [dJP00] 4.3. �

Remark 3.8. A similar statement holds for non-hypersurfaces, using insteadof the partial derivatives the k × k-minors of the Jacobian matrix of a system ofequations of X in W , with k the codimension of X in W at a [dJP00]. The choice ofthe system of defining polynomials of the variety is significant: The ideal generatedby them has to be radical, otherwise the concept of singularity has to be developedscheme-theoretically, allowing non-reduced schemes. Over non-perfect fields, thecriterion of the proposition does not hold [Zar47].

Definition 3.9. The characterization of singularities by the vanishing of thepartial derivatives is known as the Jacobian criterion for smoothness.

Corollary 3.10. The singular locus Sing(X) of X is closed in X.

12 HERWIG HAUSER

Definition 3.11. A point a of X is a plain point of X if the local ring OX,ais isomorphic to the local ring OAd,0 ' K[x1, . . . , xd](x1,...,xd) of some affine spaceAd at 0 (with d the dimension of X at a). Equivalently, there exists an openneighbourhood U of a in X which is biregularly isomorphic to an open subset V ofsome affine space Ad. The variety X is plain if all its points are plain.

Theorem 3.12. (Bodnar-Hauser-Schicho-Villamayor) The blowup of a plainvariety defined over an infinite field along a regular center is again plain.

Proof. [BHSV08] Thm. 4.3. �

Definition 3.13. A variety X is rational if it has a dense open subset Uwhich is biregularly isomorphic to a dense open subset V of some affine space Ad.Equivalently, the function field K(X) = Quot(K[X]) is isomorphic to the field ofrational functions K(x1, . . . , xd) of Ad.

Remark 3.14. A plain complex variety is smooth and rational. The converseis true for curves and surfaces, and unknown in arbitrary dimension [BHSV08].

Definition 3.15. A point a is a normal crossings point of X if the formalneighbourhood (X, a) is isomorphic to the formal neighbourhood (Y , 0) of a unionY of coordinate subspaces of An at 0. The point a is a simple normal crossingspoint of X if it is a normal crossings point and all components of X passing througha are regular at a. The variety X has normal crossings, respectively simple normalcrossings, if the property holds at all of its points.

Remark 3.16. In the case of schemes, a normal crossings scheme may benon-reduced in which case the components of the scheme Y are equipped withmultiplicities. Equivalently, Y is defined locally at 0 in An by a monomial ideal ofK[x1, . . . , xn].

Proposition 3.17. A point a is a normal crossings point of a subvariety Xof a regular ambient variety W if and only if there exists a regular system ofparameters x1, . . . , xn of OW,a so that the germ (X, a) is defined in (W,a) by aradical monomial ideal in x1, . . . , xn, i.e., each component of (X, a) is defined by asubset of the coordinates.

Remark 3.18. In the case of schemes, the monomial ideal need not be radical.

Definition 3.19. Two subvarieties X and Y of a regular ambient variety Wmeet transversally at a point a of W if they are regular at a and if the union X ∪Yhas normal crossings at a.

Remark 3.20. The definition differs from the corresponding notion in differ-ential geometry, where it is required that the tangent spaces of the two varietiesat intersection points sum up to the tangent space of the ambient variety at therespective point. In the present context, an inclusion Y ⊂ X of two regular vari-eties is considered as being transversal, and also any two coordinate subspaces inAn meet transversally. In the case of schemes, the union X ∪ Y has to be definedby the product of the defining ideals, not their intersection.

Definition 3.21. A variety X is a cylinder over a subvariety Z ⊂ X at apoint a of Z if the formal neighborhood (X, a) is isomorphic to a cartesian product(Y , a)× (Z, a) for some (positive-dimensional) subvariety Y of X which is regular


at a. One also says that X is trivial or a cylinder along Y at a with transversalsection Z.

Definition 3.22. Let X and F be varieties, and let 0 be a distinguished pointon F . The set S of points a ofX where the formal neighborhood (X, a) is isomorphicto (F , 0) is called the triviality locus of X of singularity type (F , 0).

Theorem 3.23. (Ephraim, Hauser-Muller) For any variety F and point 0 on F ,the triviality locus S ofX of singularity type (F , 0) is locally closed and regular inX,andX is a cylinder along S. Any subvariety Z ofX such that (X, a) ' (S, a)×(Z, a)is unique up to formal isomorphism at a.

Proof. [Eph78] Thm. 2.1, [HM89] Thm. 1. �

Corollary 3.24. The singular locus Sing(X) and the non-normal crossingslocus of a variety X are closed subvarieties.

Proof. [Mum99] III.4, Prop. 3, p. 170, [Bod04]. �

Definition 3.25. A variety X is a cartesian product, if there exist positivedimensional varieties Y and Z such that X is biregularly isomorphic to Y × Z.Analogous definitions hold for germs (X, a) and formal germs (X, a).

Theorem 3.26. (Hauser-Muller) Let X, Y and Z be varieties with points a,b and c on them, respectively. The formal germs of X × Z at (a, c) and Y × Z at(b, c) are isomorphic if and only if (X, a) and (Y , b) are isomorphic.

Proof. [HM90], Thm. 1. �

Example 3.27. The element x√

1 + x is a regular parameter system of K[[x]]which does not stem from a regular parameter system of the local ring K[x](x).

Example 3.28. The elements y2 − x3 − x and y form a regular parametersystem of OA2,0 but the zeroset X of y2 = x3 − x is not locally at 0 isomorphic toA1. However, (X, 0) is formally isomorphic to (A1, 0).

Example 3.29. The set of plain points of a variety X is Zariski open.

Example 3.30. Let X be the plane cubic curve defined by x3 + x2 − y2 = 0in A2. The origin 0 ∈ X is a normal crossings point, but X is not locally at 0biregularly isomorphic to the union of the two diagonals of A2.

Example 3.31. Let X be the surface in A3 defined by x2 − y2z = 0, theWhitney umbrella or pinch point singularity. The singular locus is the z-axis. Theorigin is not a normal crossings point of X. For a 6= 0 a point on the z-axis, a is anormal crossings point but not a simple normal crossings point of X. The formalneighbourhoods of X at points a 6= 0 on the z-axis are isomorphic to each other,since they are isomorphic to the formal neighbourhood at 0 of the union of twotransversal planes in A3.

Example 3.32. Prove that the non-normal crossings locus of a variety is closed.Try to find equations for it [Bod03].

Example 3.33. Find a local invariant that measures reasonably the distanceof a point a of X from being a normal crossings point.

14 HERWIG HAUSER

Example 3.34. Do the varieties defined by the following equations have normalcrossings, respectively simple normal crossings, at the origin? Vary the ground field.

(a) x2 + y2 = 0,(b) x2 − y2 = 0,(c) x2 + y2 + z2 = 0,(d) x2 + y2 + z2 + w2 = 0,(e) xy(x− y) = 0,(f) xy(x2 − y) = 0,(g) (x− y)z(z − x) = 0.

Example 3.35. Visualize the zeroset of (x− y2)(x− z)z = 0 in A3R.

Example 3.36. ∗ Formulate and then prove the theorem of local analytictriviality in positive characteristic, cf. Thm. 1 of [HM89].

Example 3.37. Find a coordinate free description of normal crossings singu-larities. Find an algorithm that tests for normal crossings [Fab11, Fab12].

Example 3.38. The singular locus of a cartesian product X × Y is the unionof Sing(X)× Y and X × Sing(Y ).

Example 3.39. Call a finite union X =⋃Xi of regular varieties mikado if

all intersections are (scheme-theoretically) regular. The intersections are definedby the sums of the ideals, not taking their radical. Find the simplest example of avariety which is not mikado but for which all pairwise intersections are non-singular.

Example 3.40. Interpret the family given by taking the germs of a varietyX at varying points a ∈ X as the germs of the fibers of a morphism of varieties,equipped with a section.

4. Lecture IV: Blowups

Blowups, also known as monoidal transformations, can be introduced in severalways. The respective equivalences will be proven in the second half of this section.All varietes are reduced but not necessarily irreducible, and subvarieties are closedif not mentioned differently. Schemes will be noetherian but not necessarily of finitetype over a field. To easen the exposition they are often assumed to be affine, i.e.,of the form X = Spec(R) for some ring R. Points of varieties are closed, pointsof schemes can also be non-closed. The coordinate ring of an affine variety X isdenoted by K[X] and the structure sheaf of a scheme X by OX , with local ringsOX,a at points a ∈ X.

References providing additional material on blowups are, among many others,[Hir64], chap. III, and [EH00], chap. IV.2.

Definition 4.1. A subvariety Z of a variety X is called a hypersurface in Xif the codimension of Z in X at any point a of Z is 1,

dima Z = dimaX − 1.

Remark 4.2. In the case where X is non-singular and irreducible, a hyper-surface is locally defined at any point a by a single non-trivial equation, i.e., anequation given by a non-zero and non-invertible element h of OX,a. This need notbe the case for singular varieties, see ex. 4.21. Hypersurfaces are a particular caseof effective Weil divisors [Har77] II, Rmk. 6.17.1, p. 145.


Definition 4.3. A subvariety Z of an irreducible variety X is called a Cartierdivisor in X at a point a ∈ Z if Z can be defined locally at a by a single equationh = 0 for some non-zero element h ∈ OX,a. If X is not assumed to be irreducible, his required to be a non-zero divisor of OX,a. This excludes the possibility that Z isa component or a union of components of X. The subvariety Z is called a Cartierdivisor in X if it is a Cartier divisor at each of its points. The empty subvariety isconsidered as a Cartier divisor. A (non-empty) Cartier divisor is a hypersurface inX, but not conversely, and its complement X \ Z is dense in X. Cartier divisorsare, in a certain sense, the largest closed and properly contained subvarieties of X.If Z is Cartier in X, the ideal I defining Z in X is called invertible or locally freeof rank 1 [Har77] II, Prop. 6.13, p. 144, [EH00] III.2.5, p. 117.

Definition 4.4. (Blowup via universal property) A variety X together with amorphism π : X → X is called a blowup of X with center a (closed) subvariety Zof X, or a blowup of X along Z, if the inverse image E = π−1(Z) of Z is a Cartierdivisor in X and π is universal with respect to this property: For any morphismτ : X ′ → X such that τ−1(Z) is a Cartier divisor in X ′, there exists a uniquemorphism σ : X ′ → X so that τ factors through σ, say τ = π ◦ σ,

X ′∃1 σ //____

τ

##HHHHHHHHH X

π

��

X

The morphism π is also called the blowup map. The subvariety E of X is a Cartierdivisor, in particular a hypersurface, and called the exceptional divisor or excep-tional locus of the blowup. One says that π contracts E to Z.

Remark 4.5. By the universal property, a blowup of X along Z is unique up tounique isomorphism. It is therefore called the blowup of X along Z. If Z is alreadya Cartier divisor in X, then X = X and π is the identity by the universal property.In particular, this is the case when X is non-singular and Z is a hypersurface in X.

Definition 4.6. The Rees algebra of an ideal I of a commutative ring R is thegraded R-algebra

Rees(I) =∞⊕i=0

Ii =∞⊕i=0

Ii · ti ⊂ R[t],

where Ii denotes the i-fold power of I, with I0 set equal to R. The variable t isgiven degree 1. Write R for Rees(I) when I is clear from the context. The Reesalgebra of I is generated by elements of degree 1, and R embeds naturally into Rby sending an element g of R to the degree 0 element g · t0. If I is finitely generatedby elements g1, . . . , gk ∈ R, then R = R[g1t, . . . , gkt]. The Rees algebras of thezero-ideal I = 0 and of the whole ring I = R equal R, respectively R[t]. If I is aprincipal ideal generated by a non-zero divisor of R, the Rees algebra is isomorphicto R. The Rees algebras of an ideal I and its k-th power Ik are isomorphic asgraded R-algebras, for any k ≥ 1.

Definition 4.7. (Blowup via Rees algebra) Let X = Spec(R) be an affinescheme and let Z = Spec(R/I) be a closed subscheme of X defined by an idealI of R. Denote by R = Rees(I) the Rees algebra of I over R, equipped with the

16 HERWIG HAUSER

induced grading. The blowup of X along Z is the scheme X = Proj(R) togetherwith the morphism π : X → X given by the natural graded ring homomorphismR → R. The subscheme E = π−1(Z) of X is called the exceptional divisor of theblowup.

Definition 4.8. (Blowup via secants) Let W = An be affine space over K andlet p be a fixed point of An. Equip An with a vector space structure by identifyingit with its tangent space TpAn. For a point a ∈ An different from p, denote byg(a) the secant line in An through p and a, considered as an element of projectivespace Pn−1 = P(TpAn). The morphism

γ : An \ {p} → Pn−1, a 7→ g(a),

is well defined. The Zariski closure X of the graph Γ of γ inside An×Pn−1 togetherwith the restriction π : X → X of the projection map An×Pn−1 → An is the pointblowup of An with center p.

Definition 4.9. (Blowup via closure of graph) Let X be an affine variety withcoordinate ring K[X] and let Z = V(I) be a subvariety of X defined by an ideal Iof K[X] generated by elements g1, . . . , gk. The morphism

γ : X \ Z → Pk−1, a 7→ (g1(a) : · · · : gk(a)),

is well defined. The Zariski closure X of the graph Γ of γ inside X ×Pk−1 togetherwith the restriction π : X → X of the projection map X ×Pk−1 → X is the blowupof X along Z. It does not depend, up to isomorphism over X, on the choice of thegenerators gi of I.

Definition 4.10. (Blowup via equations) Let X be an affine variety withcoordinate ring K[X] and let Z = V(I) be a subvariety of X defined by an idealI of K[X] generated by elements g1, . . . , gk. Assume that g1, . . . , gk form a regularsequence in K[X]. Let (u1 : . . . : uk) be projective coordinates on Pk−1. Thesubvariety X of X × Pk−1 defined by the equations

ui · gj − uj · gi = 0, i, j = 1, . . . k,

together with the restriction π : X → X of the projection map X × Pk−1 → X isthe blowup of X along Z. It does not depend, up to isomorphism over X, on thechoice of the generators gi of I.

Remark 4.11. This is a special case of the preceding definition of blowup asthe closure of a graph. If g1, . . . , gk do not form a regular sequence, the subvarietyX of X × Pk−1 may require more equations, see ex. 4.43.

Definition 4.12. (Blowup via affine charts) Let W = An be affine space overK with coordinates x1, . . . , xn. Let Z ⊂ W be a coordinate subspace defined byequations xj = 0 for j in some subset J ⊂ {1, . . . n}. Set Uj = An for j ∈ J , andglue two affine charts Uj and U` via the transition maps

xi 7→ xi/xj , if i ∈ J \ {j, `},

xj 7→ 1/x`,

x` 7→ xjx`,

xi 7→ xi, if i 6∈ J.


This yields a variety W . Define a morphism π : W → W by the chart expressionsπj : An → An for j ∈ J as follows,

xi 7→ xi, if i 6∈ J \ {j},xi 7→ xixj , if i ∈ J \ {j}.

The variety W together with the morphism π : W → W is the blowup of W = Analong Z. The map πj : An → An is called the j-th affine chart of the blowup map,or the xj-chart. It depends on the choice of coordinates.

Definition 4.13. (Blowup via ring extensions) Let I be a non-zero ideal in anoetherian integral domain R, generated by non-zero elements g1, . . . , gk of R. Theblowup of R in I is given by the ring extensions

R ↪→ Rj = R

[g1gj, . . . ,

gkgj

], j = 1, . . . , k,

inside the rings Rgj= R[ 1

gj], where the rings Rj are glued pairwise via the inclu-

sions Rgj, Rg`

⊂ Rgjg`, for j, ` = 1, . . . , k. The blowup does not depend, up to

isomorphism over X, on the choice of the generators gi of I.

Remark 4.14. The seven notions of blowup given in the preceding definitionsare all essentially equivalent. It will be convenient to prove the equivalence in thelanguage of schemes, though the substance of the proofs is inherent to varieties.

Definition 4.15. (Local blowup via germs) Let π : X ′ → X be the blowupof X along Z, and let a′ be a point of X ′ mapping to the point a ∈ X. The localblowup of X along Z at a′ is the morphism of germs π : (X ′, a′)→ (X, a). It is givenby the dual homomorphism of local rings π∗ : OX,a → OX′,a′ . This terminology isalso used for the completions of the local rings giving rise to a morphism of formalneighborhoods π : (X ′, a′)→ (X, a) with dual homomorphism π∗ : OX,a → OX′,a′ .

Definition 4.16. (Local blowup via localizations of rings) Let R = (R,m) bea local ring and let I be an ideal of R with Rees algebra R =

⊕∞i=0 I

i. Any non-zeroelement g of I defines a homogeneous element of degree 1 in R, also denoted by g.The quotient ring Rg inherits from R the structure of a graded ring. The set ofdegree 0 elements of Rg forms a ring, denoted by R[Ig−1], and consists of fractionsf/g` with f ∈ I` and ` ∈ N. The localization R[Ig−1]p of R[Ig−1] at any primeideal p of R[Ig−1] containing the maximal ideal m of R together with the naturalring homomorphism α : R→ R[Ig−1]p is called the local blowup of R with center Iassociated to g and p.

Remark 4.17. The same definition can be made for non-local rings R, givingrise to a local blowup Rq → Rq[Ig−1]p with respect to the localization Rq of R atthe prime ideal q = p ∩R of R.

Theorem 4.18. Let X = Spec(R) be an affine scheme and Z a closed sub-scheme defined by an ideal I of R. The blowup π : X → X of X along Z whendefined as Proj(R) of the Rees algebra R of I satisfies the universal property ofblowups.

Proof. (a) It suffices to prove the universal property on the affine charts ofan open covering of Proj(R), since on overlaps the local patches will agree by theiruniqueness. This in turn reduces the proof to the local situation.

18 HERWIG HAUSER

(b) Let β : R → S be a homomorphism of local rings such that β(I) · S isa principal ideal of S generated by a non-zero divisor, for some ideal I of R. Itsuffices to show that there is a unique homomorphism of local rings γ : R′ → Ssuch that R′ equals a localization R′ = R[Ig−1]p for some non-zero element g of Iand a prime ideal p of R[Ig−1] containing the maximal ideal of R, and such thatthe diagram

R

β

��

α // R′

γ~~}}

}}}}

}}

Scommutes, where α : R→ R′ denotes the local blowup of R with center I specifiedby the choice of g and p. The proof of the local statement goes in two steps.

(c) There exists an element f ∈ I such that β(f) generates β(I) · S: Let h ∈ Sbe a non-zero divisor generating β(I) · S. Write h =

∑ni=1 β(fi)si with elements

f1, . . . , fn ∈ I and s1, . . . , sn ∈ S. Write β(fi) = tih with elements ti ∈ S. Thus,h =

(∑ni=1 siti

)h. Since h is a non-zero divisor, the sum

∑ni=1 siti equals 1. In

particular, there is an index i for which ti does not belong to the maximal ideal ofS. This implies that this ti is invertible in S, so that h = t−1

i β(fi). In particular,(β(fi)) generates β(I) · S.

(d) Let f ∈ I be as in (c). By assumption, the image β(f) is a non-zerodivisor in S. For every ` ≥ 0 and every h` ∈ I`, there is an element a` ∈ Ssuch that β(h`) = a`β(f)`. Since β(f) is a non-zero divisor, a` is unique withthis property. For an arbitrary element

∑n`=0 h`/g

` of R[Ig−1] with h` ∈ I` setδ(∑n`=0 h`/g

`) =∑n`=0 a`. This defines a ring homomorphism δ : R[Ig−1]→ S that

restricts to β on R. By definition, δ is unique with this property. Let p ⊂ R[Ig−1] bethe inverse image under δ of the maximal ideal of S. This is a prime ideal of R[Ig−1]which contains the maximal ideal of R. By the universal property of localization,δ induces a homomorphism of local rings γ : R[Ig−1]p → S that restricts to β onR, i.e., satisfies γ ◦ α = β. By construction, γ is unique. �

Theorem 4.19. Let X = Spec(R) be an affine scheme and let Z a closedsubscheme defined by an ideal I of R. The blowup of X along Z defined by Proj(R)can be covered by affine charts as described in the respective definition.

Proof. For g ∈ I denote by R[Ig−1] ⊂ Rg the subring of the quotient ring Rggenerated by homogeneous elements of degree 0 of the form h/g` with h ∈ I`

and ` ∈ N. This gives an injective ring homomorphism R[Ig−1] → Rg. Letnow g1, . . . , gk be generators of I. Then X = Proj(R) is covered by the prin-cipal open sets Spec(R[Ig−1

i ]) = Spec(R[g1/gi, . . . , gk/gi]). The chart expressionSpec(R[g1/gi, . . . , gk/gi]) → Spec(R) of the blowup map π : X → X follows nowby computation. �

Theorem 4.20. Let X = Spec(R) be an affine scheme and Z a closed sub-scheme defined by the ideal I of R with generators g1, . . . , gk. The blowup of Xalong Z defined by Proj(R) of the Rees algebra equals the closure of the graph Γof γ : X \Z → Pk−1, a 7→ (g1(a) : · · · : gk(a)). If g1, . . . , gk form a regular sequence,the closure is defined as a subscheme of X × Pk−1 by equations as indicated in therespective definition.


Proof. (a) Let Uj ⊂ Pk−1 be the affine chart given by uj 6= 0, with isomor-phism Uj ' Ak−1, (u1 : . . . : uk) 7→ (u1/uj , . . . , uk/uj). The j-th chart expres-sion of γ equals a 7→ (g1(a)/gj(a), . . . , gk(a)/gj(a)) and is defined on the principalopen set gj 6= 0 of X. The closure of the graph of γ in Uj is therefore givenby Spec(R[g1/gi, . . . , gk/gi]). The preceding theorem then establishes the requiredequality.

(b) If g1, . . . , gk form a regular sequence, their only linear relations over R arethe trivial ones, so that

R[Ig−1i ] = R[g1/gi, . . . gk/gi] ' R[t1, . . . , tk]/(gitj − gj , j = 1, . . . , k).

The i-th chart expression of π : X → X is then given by the ring inclusion R →R[t1, . . . , tk]/(gitj − gj , j = 1, . . . , k). �

Example 4.21. For the cone X in A3 of equation x2 + y2 = z2, the line Y inA3 defined by x = y − z = 0 is a hypersurface of X at each point. It is a Cartierdivisor at any point a ∈ Y \ {0} but it is not a Cartier divisor at 0. The doubleline Y ′ in A3 defined by x2 = y − z = 0 is a Cartier divisor of X since it can bedefined in X by y − z = 0. The subvariety Y ′′ in A3 consisting of the two linesdefined by x = y2− z2 = 0 is a Cartier divisor of X since it can be defined in X byx = 0.

Example 4.22. For the surface X : x2y−z2 = 0 in A3, the subvariety Y definedby y2 − xz = x3 − yz = 0 is the singular curve parametrized by t 7→ (t3, t4, t5).It is everywhere a Cartier divisor except at 0: there, the local ring of Y in X isK[x, y, z](x,y,z)/(x2y − z2, y2 − xz, x3 − yz), which defines a singular cubic. It isof codimension 1 in X but a non-complete intersection. Thus it is not a Cartierdivisor. At any other point a = (t3, t4, t5), t 6= 0 of Y , one has OY,a = S/(x2y −z2, y2/x − z, x3 − yz) = S/(x2y − z2, y2/x − z) with S = K[x, y, z](x−t3,y−t4,z−t5)the localization of K[x, y, z] at a. Hence Y is Cartier divisor there.

Example 4.23. Let Z be one of the axes of the cross X : xy = 0 in A2, e.g.,the x-axis. At any point a on the y-axis except the origin, Z is Cartier: one hasOX,a = K[x, y](x,y−a)/(xy) = K[x, y](x,y−a)/(x) and the element h = y defining Z isa unit in this local ring. At the origin 0 of A2 the local ring of X is K[x, y](x,y)/(xy)and Z is locally defined by h = y which is a zero-divisor in OX,0. Thus Z is notCartier in X at 0.

Example 4.24. Let Z be the origin of X = A2. Then Z is not a hypersurfaceand hence not Cartier in X.

Example 4.25. The origin Z of X = V (xy, x2) in A2 is not a Cartier divisorin X.

Example 4.26. Are Z = V (x2) in A1 and Z = V (x2y) in X = A2 Cartierdivisors?

Example 4.27. Let Z be the reduced origin of X = V (xy, x2) in A2. Showthat Z is not a Cartier divisor in X.

Example 4.28. Let X = A1 be the affine line with coordinate ring R = K[x]and let Z = 0 be the origin. The Rees algebra R = K[x, xt] ⊂ K[x, t] with respectto Z is isomorphic, as a graded ring, to a polynomial ring K[u, v] in two variables

20 HERWIG HAUSER

with deg u = 0 and deg v = 1. Therefore, the point blowup X of X is isomorphicto A1 × P0 = A1, and π : X → X is the identity.

More generally, let X = An be n-dimensional affine space with coordinatering R = K[x1, . . . , xn]and let Z ⊂ X be a hypersurface defined by some non-zerog ∈ K[x1, . . . , xn]. The Rees algebra R = K[x1, . . . , xn, gt] ⊂ K[x1, . . . , xn, t] withrespect to Z is isomorphic, as a graded ring, to a polynomial ring K[u1, . . . , un, v]in n + 1 variables with deg ui = 0 and deg v = 1. Therefore the blowup X of Xalong Z is isomorphic to An × P0 = An, and π : X → X is the identity.

Example 4.29. Let X = Spec(R) be an affine scheme and Z ⊂ X be definedby some non-zero-divisor g ⊂ R. The Rees algebra R = R[gt] ⊂ R[t] with respect toZ is isomorphic, as a graded ring, to R[v] with deg r = 0 for r ∈ R and deg v = 1.Therefore the blowup X of X along Z is isomorphic to X × Proj0 = X, andπ : X → X is the identity.

Example 4.30. Take X = V (xy) ⊂ A2 with coordinate ring R = K[x, y]/(xy).Let Z be defined in X by y = 0. The Rees algebra of R with respect to Z equalsR = R[yt] ∼= K[x, y, u]/(xy, xu) with deg x = deg y = 0 and deg u = 1.

Example 4.31. Let X = Spec(R) be an affine scheme and let Z ⊂ X bedefined by some zero-divisor g 6= 0 in R, say h · g = 0 for some non-zero h ∈ R.The Rees algebra R = R[gt] ⊂ R[t] with respect to Z is isomorphic, as a gradedring, to R[v]/(h · v) with deg r = 0 for r ∈ R and deg v = 1. Therefore, the blowupX of X along Z equals the closed subvariety of X × P0 = X defined by h · v = 0,and π : X → X is the inclusion map.

Example 4.32. Take in the situation of the preceding example X = V (xy) ⊂A2 and g = y, h = x. Then R = K[x, y]/(xy) and R = R[yt] ' K[x, y, u]/(xy, xu)with deg x = deg y = 0 and deg u = 1.

Example 4.33. Let X = Spec(R) be an affine scheme and let Z ⊂ X bedefined by some nilpotent element g 6= 0 in R, say gk = 0 for some k ≥ 1. TheRees algebra R = R[gt] ⊂ R[t] with respect to Z is isomorphic, as a graded ring,to R[v]/(vk) with deg r = 0 for r ∈ R and deg v = 1. The blowup X of X is theclosed subvariety of X × P0 defined by vk = 0.

Example 4.34. Let X = A2 and let Z be defined in X by I = (x, y). TheRees algebra R = K[x, y, xt, yt] ⊂ K[x, y, t] of R = K[x, y] with respect to Zis isomorphic, as a graded ring, to the factor ring K[x, y, u, v]/(xv − yu), withdeg x = deg y = 0 and deg v = deg u = 1. It follows that the blowup X of Xalong Z embeds naturally as the closed and regular subvariety of A2 × P1 definedby xv − yu = 0, the morphism π : X → X being given by the restriction to X ofthe first projection A2 × P1 → A2.

Example 4.35. Let X = A2 and let Z be defined in X by I = (x, y2). TheRees algebra R = K[x, y, xt, y2t] ⊂ K[x, y, t] of R = K[x, y] with respect to Zis isomorphic, as a graded ring, to the factor ring K[x, y, u, v]/(xv − y2u), withdeg x = deg y = 0 and deg u = deg v = 1. It follows that the blowup X of X alongZ embeds naturally as the closed and singular subvariety of A2 × P1 defined byxv − y2u = 0, the morphism π : X → X being given by the restriction to X of thefirst projection A2 × P1 → A2.


Example 4.36. Let X = A3 and let Z be defined in X by I = (xy, z). TheRees algebra R = K[x, y, z, xyt, zt] ⊂ K[x, y, z, t] of R = K[x, y, z] with respectto Z is isomorphic, as a graded ring, to K[u, v, w, r, s]/(uvs − wr), with deg u =deg v = degw = 0 and deg r = deg s = 1. It follows that the blowup X of X alongZ embeds naturally as the closed and singular subvariety of A3 × P1 defined byuvs−wr = 0, the morphism π : X → X being given by the restriction to X of thefirst projection A3 × P1 → A3.

Example 4.37. Let X = A1Z be the affine line over the integers and let Z

be defined in X by I = (x, p) for a prime p ∈ Z. The Rees algebra R =Z[x, xt, pt] ⊂ Z[x, t] of R = Z[x] with respect to Z is isomorphic, as a gradedring, to Z[u, v, w]/(xw− pv), with deg u = 0 and deg v = degw = 1. It follows thatthe blowup X of X along Z embeds naturally as the closed and regular subvarietyof A1

Z × P1Z defined by xw − pv, the morphism π : X → X being given by the

restriction to X of the first projection A1Z × P1

Z → A1Z.

Example 4.38. Let X = A2Z20

with Z20 = Z/20Z, and let Z be defined in X

by I = (x, 2y). The Rees algebra R = Z20[x, y, xt, 2yt] ⊂ Z20[x, y, t] of R = Z20[x]with respect to Z is isomorphic, as a graded ring, to Z20[u, v, w, z]/(xz−2vw), withdeg u = deg v = 0 and degw = deg z = 1.

Example 4.39. Let R = Z6 with Z6 = Z/6Z, and let Z be defined in X

by I = (2). The Rees algebra R = Z6[2t] ⊂ Z6[t] of R = with respect to Z isisomorphic, as a graded ring, to Z3[u] with deg u = 1.

Example 4.40. Let R = K[[x, y]] be a formal power series ring in two variablesx and y, and let Z be defined in X by I be the ideal generated by ex − 1 andln(y). The Rees algebra R of R with respect to Z equals R[(ex − 1)t, ln(y)t] =K[[x, y]][(ex−1)t, ln(y)t] ∼= K[[x, y]][u, v]/(ln(y)u−(ex−1)v) with deg u = deg v = 1.

Example 4.41. Let X = An. The point blowup X ⊂ X × Pn−1 of X at theorigin is defined by the ideal (uixj−ujxi, i, j = 1, . . . n) in K[x1, . . . , xn, u1, . . . , un].This is a graded ring, where deg xi = 0 and deg ui = 1 for all i = 1, . . . , n. But thering

K[x1, . . . , xn, u1, . . . , un]/(uixj − ujxi, i, j = 1, . . . n)

is isomorphic, as a graded ring, to R = K[x1, . . . , xn, x1t, . . . , xnt] ⊂ K[x1, . . . , xn, t]with deg xi = 0, deg t = 1. The ring R is the Rees-algebra of the ideal (x1, . . . , xn)of K[x1, . . . , xn]. Thus X is isomorphic to Proj

(⊕d≥0(x1, . . . , xn)d

).

Example 4.42. The i-th affine chart of the blowup An of An in the idealI = (x1, . . . , xk) is isomorphic to An, for i = 1, . . . , k, via the ring isomorphism

K[x1, . . . , xn] ' K[x1, . . . , xn, t1, . . . , ti, . . . tk]/(xitj − xj , j = 1, . . . , k, j 6= i),

where xj 7→ tj for j = 1, . . . , k, j 6= i, respectively xj 7→ xj for j = k + 1, . . . , nand j = i. The inverse is given by xj 7→ xixj for j = 1, . . . , k, j 6= i, respectivelyxj 7→ xj for j = k + 1, . . . , n and j = i, respectively tj 7→ xj for j = 1, . . . , k, j 6= i.

Example 4.43. Let X = A2 be the affine plane and let g1 = x2, g2 = xy, g3 =y3 be generators of the ideal I ⊂ K[x, y]. The gi do not form a regular sequence.The subvariety of A2×P2 defined by all giuj − gjui = 0 is singular, but the blowupof A2 in I is regular.

22 HERWIG HAUSER

Example 4.44. Compute the following blowups:(a) A2 in the center (x, y)(x, y2),(b) A3 in the centers (x, yz) and (x, yz)(x, y)(x, z),(c) A3 in the center (x2 + y2 − 1, z).(d) The plane curve x2 = y in the origin.Use affine charts and ring extensions to determine at which points the resulting

varieties are regular or singular.

Example 4.45. Blow up A3 in 0 and compute the inverse image of x2+y2 = z2.

Example 4.46. Blow up A2 in the point (0, 1). What is the inverse image ofthe lines x+ y = 0 and x+ y = 1?

Example 4.47. Compute the chart transition map for the blowup of A3 in thez-axis.

Example 4.48. Blow up the cone x2 + y2 = z2 in one of its lines.

Example 4.49. Blow up A3 in the circle x2 + (y + 2)2 − 1 = z = 0 and in theelliptic curve y2 − x3 − x = z = 0.

Example 4.50. Interpret the blowup of A3 in (x, yz)(x, y)(x, z) as a composi-tion of blowups in regular centers [Hau00, FW11, Lev01].

Example 4.51. Show that the blowup of An along a coordinate subspace Zequals the cartesian product of the point-blowup in a transversal subspace V of Anof complementary dimension (with respect to Z) with the identity map on Z.

Example 4.52. Show that the ideals (x1, . . . , xn) and (x1, . . . , xn)m define thesame blowup of An when taken as center.

Example 4.53. Let E be a normal crossings subvariety of An and let Z be asubvariety of An such that E∪Z also has normal crossings (the union being definedby the product of ideals). Show that the inverse image of E under the blowup ofAn along Z has again normal crossings. Show by an example that the assumptionon Z cannot be dropped in general.

Example 4.54. Draw a real picture of the blowup of A2 at the origin.

Example 4.55. Show that the blowup An of An with center a point is non-singular.

Example 4.56. Describe the geometric construction of the blowup A3 of A3

with center 0. What is π−1(0)? For a chosen affine chart W ′ of A3, consider allcylinders Y over a circle in W ′ centered at the origin and parallel to a coordinateaxis. What is the image of Y under π in A3?

Example 4.57. Compute the blowup of the Whitney umbrella X = V (x2 −y2z) ⊂ A3 with center 0 and the three coordinate axes respectively.

Example 4.58. Determine the locus of points of the Whitney umbrella X =V (x2−y2z) where the singularities are normal crossings, respectively simple normalcrossings. Blow up the complements of these loci and compare with the precedingexample [Kol07] ex. 3.6.1, p. 123, [BDMP12] Thm. 3.4.


Example 4.59. Let Z be a regular center in An, with induced blowup π :An → An, and let a′ be a point of An mapping to a point a ∈ Z. Show that it ispossible to choose local formal coordinates at a, i.e., a regular parameter system ofOAn,a, so that the center is a coordinate subspace, and so that a′ the origin of oneof the affine charts of An. Is this also possible with a regular parameter system ofOAn,a?

Example 4.60. Let X = V (f) be a hypersurface in An and Z a regular closedsubvariety which is contained in the locus S of points of X where f attains itsmaximal order. Let π : An → An be the blowup along Z and let Xs = V (fs)be the strict transform of X, defined as the Zariski closure of π−1(X \ Z) in An.Show that for points a ∈ Z and a′ ∈ E = π−1(Z) with π(a′) = a the inequalityorda′f ′ ≤ ordaf holds. Do the same for subvarieties defined by arbitrary ideals.

Example 4.61. Find an example of a variety X for which the dimension ofthe singular locus increases under the blowup of a closed regular center Z that iscontained in the top locus of X [Hau98] ex. 9.

Example 4.62. Let π : (An, a′) → (An, a) be the local blowup of An withcenter the point a, considered at a point a′ ∈ E. Let be given a local coordinatesx1, . . . , xn at a. Determine the coordinate changes in (An, a) which make the chartexpression of π monomial.

Example 4.63. Let x1, . . . , xn be local coordinates at a so that the local blowupπ : (An, a′) → (An, a) is monomial with respect to them. Determine the formalautomorphisms of (An, a′) which commute with the local blowup.

Example 4.64. Compute the blowup of X = Spec(Z[x]) in the ideals (x, p)and (x, pq) where p and q are primes.

Example 4.65. The zeroset in A3 of the non-reduced ideal (x, yz)(x, y)(x, z) =(x3, x2y, x2z, xyz, y2z) is the union of the y- and the z-axis. Taken as center, theresulting blowup of A3 equals the composition of two blowups: The first blowup hascenter the ideal (x, yz) in A3, giving a three-fold W1 in a regular four-dimensionalambient variety with one singular point of local equation xy = zw. The secondblowup is the point blowup of W1 with center this singular point [Hau00] Prop. 3.5.

Example 4.66. Let R := K[x, y, z] be the polynomial ring in three variablesand let I = (x, yz) ⊂ R. The blowup of R along I corresponds to the ring exten-sions:

R ↪→ R[yzx

]∼= K[s, t, u, v]/(sv − tu),

R ↪→ R

[x

yz

]∼= K[s, t, u, v]/(s− tuv).

The first ring extension defines a singular variety while the second one defines anon-singular one. Let W = A3 and Z = V (I) ⊂ W be the affine space and thesubvariety defined by I. The blowup of R along I coincides with the blowup W ′ ofW along Z.

(a) Compute the chart expressions of the blowup maps.(b) Determine the exceptional divisor.(c) Apply one more blowup to W ′ to get a non-singular variety W ′′.

24 HERWIG HAUSER

(d) Express the composition of the two blowups as a single blowup in a properlychosen ideal.

(e) Blow up A3 along the three coordinate axes. Show that the resulting varietyis non-singular.

(f) Show the same for the blowup of An along the n coordinate axes.

Example 4.67. * Consider the blowup An of An in a monomial ideal I ofK[x, . . . , xn]. Show that An may be singular. What types of singularities willoccur? Find a natural saturation procedure I I so that the blowup of An in I isregular and equal to a (natural) resolution of the singularities of An [FW11].

Example 4.68. The Nash modification of a subvariety X of An is the closureof the graph of the map which associates to each non-singular point its tangentspace, taken as an element of the Grassmanian of d-dimensional linear subspacesof Kn, where d = dim(X). For a hypersurface X defined in An by f = 0, the Nashmodification coincides with the blowup of X in the Jacobian ideal of f generatedby the partial derivatives of f .

5. Lecture V. Properties of Blowup

Proposition 5.1. Let π : X → X be the blowup of X along a subvariety Z,and let ϕ : Y → X be a morphism, the base change. Denote by p : X ×X Y → Ythe projection from the fibre product to the second factor. Let S = ϕ−1(Z) ⊂ Y

be the inverse image of Z under ϕ, and let Y be the Zariski closure of p−1(Y \ S)in X ×X Y . The restriction τ : Y → Z of p to Y equals the blowup of Y along S.

F� � //

AA

AAAA

AA Y� � //

τ

##HHHHHHHHHH X ×X Y

q//

p

��

X

π

��

S� � // Y

ϕ// X

Proof. To show that F = τ−1(S) is Cartier in Y , consider the projectionq : X ×X Y → X onto the first factor. By the commutativity of the diagram,

F = p−1(S) = p−1 ◦ ϕ−1(Z) = q−1 ◦ π−1(Z) = q−1(E).

As E Cartier in X and q is a projection, F is locally defined by a principal ideal.The associated primes of Y are the associated primes of Y not containing the idealof S. Thus, the local defining equation of E in X cannot pull back to a zero divisoron Y . This proves that F is Cartier.

To show that τ : Y → Y fulfills the universal property of blowups, let ψ : Y ′ →Y be a morphism such that ψ−1(S) is a Cartier divisor in Y ′.

X

π

��>>

>>>>

>>

Y ′

ρ//

σ //

ψ

//

X ×X Y

q

;;wwwwwwwww

p

##GGGGGGGGG

X

Y

ϕ??��

Since ψ−1(S) = ψ−1(ϕ−1(Z)) = (ϕ ◦ ψ)−1(Z), there exists by the universalproperty of the blowup π : X → X a unique map ρ : Y ′ → X such that ϕ ◦ ψ =


π ◦ ρ. By the universal property of fibre products, there exists a unique mapσ : Y ′ → X ×X Y such that q ◦ σ = ρ and p ◦ σ = ψ.

It remains to show that σ(Y ′) lies in Y ⊂ X ×X Y . Since ψ−1(S) is Cartier inY ′, its complement Y ′ \ ψ−1(S) is dense in Y ′. From

Y ′ \ ψ−1(S) = ψ−1(Y \ S) = (p ◦ σ)−1(Y \ S) = σ−1(p−1(Y \ S))

follows that σ(Y ′ \ ψ−1(S)) ⊂ p−1(Y \ S). But Y is the closure of p−1(Y \ S) inX ×X Y , so that σ(Y ′) ⊂ Y as required. �

Corollary 5.2. (a) Let π : X ′ → X be the blowup of X along a subvarietyZ, and let Y be a closed subvariety of X. Denote by Y ′ the Zariski closure ofπ−1(Y \ Z) in X ′. The restriction τ : Y ′ → Y of π to Y ′ is the blowup of Y alongY ∩ Z. In particular, if Z ⊂ Y , then τ is the blowup of Y along Z.

(b) Let U ⊂ X be an open subvariety, and let Z ⊂ X be a closed subvariety,so that U ∩ Z is closed in U . Let π : X ′ → X be the blowup of X along Z. Theblowup of U along U ∩ Z equals the restriction of π to U ′ = π−1(U).

(c) Let a ∈ X be a point. Write (X, a) for the germ of X at a, and (X, a) forthe formal neighbourhood. There are natural maps

(X, a)→ X and (X, a)→ X

corresponding to the localization and completion homomorphisms OX → OX,a →OX,a. Take a point a′ above a in the blowupX ′ ofX along a subvariety Z containinga. This gives local blowups of germs and formal neighborhoods

πa′ : (X ′, a′)→ (X, a), πa′ : (X ′, a′)→ (X, a).

The blowup of a local ring is not local in general; to get a local blowup one needsto localize also on X ′.

(d) If X1 → X is an isomorphism between varieties sending a subvariety Z1 toZ, the blowup X ′1 of X1 along Z1 is canonically isomorphic to the blowup X ′ of Xalong Z. This also holds for local isomorphisms.

(e) If X = Z × Y is a cartesian product of two varieties, and a is a given pointof Y , the blowup π : X ′ → X of X along Z × {a} is isomorphic to the cartesianproduct IdZ× τ : Z×Y ′ → Z×Y of the identity on Z with the blowup τ : Y ′ → Yof Y in a.

Proposition 5.3. Let π : X ′ → X be the blowup of X along a regular subva-riety Z with exceptional divisor E. Let Y be a subvariety of X, and Y ∗ = π−1(Y )its preimage under π. Let Y ′ be the Zariski closure of π−1(Y \ Z) in X ′. If Y istransversal to Z, i.e., Y ∪ Z has normal crossings at all points of the intersectionY ∩ Z, also Y ∗ has normal crossings at all points of Y ∗ ∩ E. In particular, if Y isregular and transversal to Z, also Y ′ is regular and transversal to E.

Proof. Having normal crossings is defined locally at each point through thecompletions of local rings. The assertion is proven by a computation in local coor-dinates for which the blowup is monomial, cf. Prop. 5.4 below. �

Proposition 5.4. Let W be a regular variety of dimension n with a regularsubvariety Z of codimension k. Let π : W ′ →W denote the blowup of W along Z,with exceptional divisor E. Let V be a regular hypersurface in W containing Z,let D be a (not necessarily reduced) normal crossings divisor in W having normal

26 HERWIG HAUSER

crossings with V . Let a be a point of V ∩ Z and let a′ ∈ E be a point lying abovea. There exist local coordinates x1, . . . , xn of W at a such that

(1) a has components a = (0, . . . , 0).(2) V is defined in W by xn−k+1 = 0.(3) Z is defined in W by xn−k+1 = . . . = xn = 0.(4) D ∩ V is defined in V locally at a by a monomial xq11 · · ·xqn

n , for someq = (q1, . . . , qn) ∈ Nn with qn−k+1 = 0.

(5) The point a′ lies in the xn-chart of W ′. The chart expression of π in thexn-chart is of the form

xi 7→ xi for i ≤ n− k and i = n,

xi 7→ xixn for n− k + 1 ≤ i ≤ n− 1,(6) In the induced coordinates of the xn-chart, the point a′ has components

a′ = (0, . . . , 0, a′n−k+2, . . . , a′n−k+d, 0, . . . , 0) with non-zero entries a′j ∈ K for n −

k + 2 ≤ j ≤ n − k + d, where d is the number of components of D whose stricttransforms do not pass through a′.

(7) The strict transform (def. 6.2) V s of V in W ′ is given in the inducedcoordinates locally at a′ by xn−k+1 = 0.

(8) The local coordinate change ϕ in W at a given by ϕ(xi) = xi+a′i ·xn makesthe local blowup π : (W ′, a′) → (W,a) monomial. It preserves the defining idealsof Z and V in W .

(9) If condition (4) is not imposed, the coordinates x1, . . . , xn at a can be chosenwith (1) to (3) and so that a′ is the origin of the xn-chart.

Proof. [Hau10b]. �

Theorem 5.5. Any projective birational morphism π : X ′ → X is a blowup ofX in an ideal I.

Proof. [Har77] Thm. 7.14. �

Example 5.6. Let X be a normal crossings subvariety of An and Z a regularclosed subscheme which is transversal to X. Show that the blowup X ′ of X alongZ is again a normal crossings scheme.

Example 5.7. * Prove that plain varieties remain plain under blowup in regularcenters [BHSV08] Thm. 4.3.

Example 5.8. * Is any rational and regular variety plain?

Example 5.9. Consider the blowup π : W ′ → W of a regular variety Walong a closed subvariety Z. Show that the exceptional divisor E = π−1(Z) is ahypersurface in W ′.

Example 5.10. The composition of two blowups W ′′ →W ′ and W ′ →W is ablowup of W in a suitable center [Bod03].

Example 5.11. A fractional ideal I over an integral domainR is anR-submoduleof Quot(R) such that rI ⊂ R for some non-zero element r ∈ R. Blowups can bedefined via Proj also for centers which are fractional ideals [Gro61]. Let I and J betwo (ordinary) non-zero ideals of R. The blowup of R along I is isomorphic to theblowup of R along J if and only if there exist positive integers k, ` and fractionalideals K, L over R such that JK = Ik and IL = Jl [Moo01] Cor. 2.


Example 5.12. For the surface x2y − z2 = 0 in A3, the subvariety Z definedby y2 − xz = x3 − yz = 0 is the singular curve parametrized by t → (t3, t4, t5). Itis everywhere Cartier except at 0. Prove that Z is not Cartier at 0.

Example 5.13. Determine the equations of the blowup X ′ in X×P2 of X = A3

along the image Z of the monomial curve (t3, t4, t5) of equations g1 = y2 − xz,g2 = yz − x3, g3 = z2 − x2y.

Example 5.14. Show that the blowup of the cone X = V (x2−yz) in A3 alongthe line Z = V (x, y) is an isomorphism locally at all points outside 0, but notglobally on X.

Example 5.15. Blow up the non-reduced point X = V (x2) in A2 in the (re-duced) origin Z = 0.

Example 5.16. Blow up the subscheme X = V (x2, xy) of A2 in the reducedorigin.

Example 5.17. Blow up the subvariety X = V (xz, yz) of A3 first in the origin,then in the x-axis, and determine the points where the resulting morphisms are localisomorphisms.

Example 5.18. The blowups of A3 along the union of the x- with the y-axis,respectively along the cusp, with ideals (xy, z) and (x3 − y2, z), are singular.

6. Lecture VI: Transforms of Ideals and Varieties under Blowup

Throughout this section, π : W ′ → W denotes the blowup of a variety W along aclosed subvariety Z, with exceptional divisor E = π−1(Z) defined by the principalideal IE of OW ′ . Let π∗ : OW → OW ′ be the dual homomorphism of π. Let X ⊂Wbe a closed subvariety, and let I be an ideal on W .

Definition 6.1. The inverse image X∗ = π−1(X) of X and the extensionI∗ = π∗(I) = I · OW ′ of I are called the total transform of X and I under π. Forf ∈ OW , denote by f∗ its image π∗(f) in OW ′ . The ideal I∗ is generated by alltransforms f∗ for f varying in I. If I is the ideal defining X in W , the ideal I∗

defines X∗ in W ′. In particular, the total transform of the center Z equals theexceptional divisor E, and W ∗ = W ′. The total transform can be non-reducedwhen considered as a subscheme of W ′.

Definition 6.2. The Zariski closure of π−1(X \ Z) in W ′ is called the stricttransform of X under π and denoted by Xs, also known as the proper or birationaltransform. The strict transform is a closed subvariety of the total transform X∗.The difference X∗ \ X ′ is contained in the exceptional locus E. If the center Zequals whole X, the strict transform is empty.

Let IU denote the restriction of an ideal I to the open set U = W \ Z. SetU ′ = π−1(U) ⊂ W ′ and let τ : U ′ → U be the restriction of π to U ′. The stricttransform Is of the ideal I is defined as τ∗(IU ) ∩ OW ′ . If the ideal I defines X inW , the ideal Is defines Xs in W ′. It equals the union of colon ideals

Is =⋃i≥0

I∗ : IiE .

Let h be an element of OW ′,a′ defining E locally in W ′ at a point a′. Then, locallyat a′,

Is = (fs, f ∈ I),

28 HERWIG HAUSER

where the strict transform fs of f is defined at a′ and up to multiplication byinvertible elements in OW ′,a′ through f∗ = hk · fs with maximal exponent k. Thevalue of k is the order of f∗ along E, i.e., the maximal number k such that f∗ ∈ IkE .By abuse of notation this is written as fs = h−k · f∗ = h−ordZf · f∗.

Lemma 6.3. Let f : R→ S be a ring homomorphism, I an ideal of R and s anelement of R, with induced ring homomorphism fs : Rs → Sf(s). Let Ie = f(I) · Sand (I ·Rs)e = fs(I ·Rs) ·Sf(s) denote the respective extensions of the ideals. Then⋃i≥0 I

e : f(s)i = (I ·Rs)e ∩ S.

Proof. Let u ∈ S. Then u ∈⋃i≥0 I

e : f(s)i if and only if uf(s)i ∈ Ie forsome i ≥ 0, say uf(s)i =

∑j ajf(xj) for elements xj ∈ I and aj ∈ S. Rewrite this

as u =∑j ajf(xj

si ). This just means that u ∈ (I ·Rs)e. �

Remark 6.4. If I is generated locally by elements f1, . . . , fk of OW , then Is

contains the ideal generated by the strict transforms fs1 , . . . , fsk of f1, . . . , fk, but

the inclusion can be strict, see the examples below.

Definition 6.5. Let K[x] = K[x1, . . . , xn] be the polynomial ring over K,considered with the natural grading given by the degree. Denote by in(g) thehomogeneous form of lowest degree of a non-zero polynomial g of K[x], called theinitial form of g. Set in(0) = 0. For a non-zero ideal I, denote by in(I) the idealgenerated by all initial forms in(g) of elements g of I, called the initial ideal of I.Elements g1, . . . , gk of an ideal I of K[x] are a Macaulay basis of I if their initialforms in(g1), . . . , in(gk) generate in(I). In [Hir64] III.1, def. 3, p. 208, such a basiswas called a standard basis, which is now used for a slightly more specific concept.By noetherianity of K[x], any ideal possesses a Macaulay basis.

Proposition 6.6. (Hironaka) The strict transform of an ideal under blowup ina regular center is generated by the strict transforms of the elements of a Macaulaybasis of the ideal.

Proof. ([Hir64], III.2, Lem. 6, p. 216, and III.6, Thm. 5, p. 238) If I ⊂ J aretwo ideals of K[[x]] such that in(I) = in(J) then they are equal, I = J . This holds atleast for degree compatible monomial orders, due to the Grauert-Hironaka-Galligodivision theorem. Therefore it has to be shown that in(gs1), . . . , in(gk) generatein(Is). But in(Is) = (in(I))s, and the assertion follows. �

Remark 6.7. The strict transform of a Macaulay basis at a point a′ of W ′ needno longer be a Macaulay basis. This is however the case if the Macaulay basis isreduced and the sequence of its orders has remained constant at a′, cf. [Hir64] III.8,Lem. 20, p. 254. More generally, taking on K[x] a grading so that all homogeneouspieces are one-dimensional and generated by monomials (i.e., a grading induced bya monomial order on Nn), the initial form of a polynomial and the initial ideal areboth monomial. In this case Macaulay bases are called standard bases. A standardbasis g1, . . . , gk is reduced if no monomial of the tails gi − in(gi) belongs to in(I).If the monomial order is degree compatible, i.e., the induced grading a refinementof the natural grading of K[x] by degree, the strict transforms of a standard basisof I generate the strict transform of the ideal.

Definition 6.8. Let I be an ideal on W and let c ≥ 0 be a natural numberless than or equal to the order d of I along the center Z. Then I∗ has order ≥ c


along E. There exists a unique ideal I ! of OW ′ such that I∗ = IdE · I !, called thecontrolled transform of I with respect to the control c. It is not defined for valuesof c > d. In case that c = d attains the maximal value, I ! is denoted by Ig andcalled the weak transform of I. It is written as Ig = I−dE · I∗ = I−ordZI

E · I∗.

Remark 6.9. The inclusions I∗ ⊂ Ig ⊂ I ! ⊂ Is are obvious. The componentsof V (Ig) which are not contained in V (Is) lie entirely in the exceptional divisorE, but can be strictly contained. For principal ideals, Ig and Is coincide. Whenthe transforms are defined scheme-theoretically, the reduction X∗red of the totaltransform X∗ of X consists of the union of E with the strict transform Xs.

Definition 6.10. A local flag F on W at a is a chain F0 = {a} ⊂ F1 ⊂ . . . ⊂Fn = W of regular closed subvarieties, respectively subschemes, Fi of dimension iof an open neighbourhood U of a in W . Local coordinates x1, . . . , xn on W at aare called subordinate to the flag F if Fi = V (xi+1, . . . , xn) locally at a. The flagF at a is transversal to a regular subvariety Z of W if each Fi is transversal to Zat a [Hau04, Pan06].

Proposition 6.11. Let π : W ′ → W be the blowup of W along a center Ztransversal to a flag F at a ∈ Z. Let x1, . . . , xn be local coordinates on W ata subordinate to F . At each point a′ of E above a there exists a unique localflag F ′ such that the coordinates x′1, . . . , x

′n on W ′ at a′ induced by x1, . . . , xn are

subordinate to F ′ [Hau04], Thm. 1.

Definition 6.12. The flag F ′ is called the transform of F under π.

Proof. It suffices to define F ′i at a′ by x′i+1, . . . , x′n. For point blowups in W ,

the transform of F is defined as follows. The point a′ ∈ E is determined by a lineL in the tangent space TaW of W at a. Let k ≤ n be the minimal index for whichTaFk contains L. For i < k, choose a regular (i+ 1)-dimensional subvariety Hi ofW with tangent space L + TaFi at a. In particular, TaHk−1 = TaFk. Let Hs

i bethe strict transform of Hi in W ′. Then set

F ′i = E ∩Hsi for i < k,

F ′i = F si for i ≥ k,to get the required flag F ′ at a′ . �

Example 6.13. Blow up A2 in 0 and compute the inverse image of the zerosetsof x2 + y2 = 0, xy = 0 and x(x− y2) = 0, as well as their strict transforms.

Example 6.14. Compute the strict transform of X = V (x2−y3, xy−z3) ⊂ A3

under the blowup of the origin.

Example 6.15. Determine the total, weak and strict transform of X = V (x2−y3, z3) ⊂ A3 under the blowup of the origin. Clarify the algebraic and geometricdifferences between them.

Example 6.16. Blow up A3 along the curve y2 − x3 + x = z = 0 and computethe strict transform of the lines x = z = 0 and y = z = 0.

Example 6.17. Let I1 and I2 be ideals of K[x] of order c1 and c2 at 0. Takethe blowup of An at zero. Show that the weak transform of Ic21 + Ic12 is the sum ofthe weak transforms of Ic21 and Ic12 .

30 HERWIG HAUSER

Example 6.18. For dim(W ) = 3, a flag at a point a consists of a regular curveF1 through a and contained in a regular surface F2. Blow up the point a in W ,so that E ' P2 is the projective plane. At the intersection point p1 of the stricttransform C1 = F s1 of F1 with E, the transformed flag F ′ is given by F s1 ⊂ F s2 .Along the intersection C2 of F s2 with E, the flag F ′ is given at each point p2 differentfrom p1 by C2 ⊂ F s2 . At any point p3 not on C2 the flag F ′ is given by C3 ⊂ E,where C3 is the projective line in E through p1 and p3.

Example 6.19. Blowing up a regular curve Z in W transversal to F thereoccur six possible configurations of Z with respect to F . Denoting by L the planein the tangent space TaW of W at a corresponding to the point a′ in E above a,these are: (1) Z = F1 and L = TaF2, (2) Z = F1 and L 6= TaF2, (3) Z 6= F1,Z ⊂ F2 and L = TaF2, (4) Z 6= F1, Z ⊂ F2 and L 6= TaF2, (5) Z 6⊂ F2 andTaF1 ⊂ L, (6) Z 6⊂ F2 and TaF1 6⊂ L. Determine in each case the flag F ′.

Example 6.20. Let An → An be the blowup of An in 0 and let a′ be theorigin of the xn-chart of An. Compute the total and strict transforms g∗ and gs forg = xd1 + . . .+ xdn−1 + xen for e = d, 2d− 1, 2d, 2d+ 1 and g =

∏i6=j(xi − xj).

Example 6.21. Determine the total, weak and strict transform of X = V (x2−y3, z3) ⊂ A3 under the blowup of A3 at the origin. Point out the geometric differ-ences between the three types of transforms.

Example 6.22. Compute the strict transform of the ideal (x2 − y3, xy − z3)under the blowup of A3 at the origin.

Example 6.23. The inclusion Ig ⊂ Is can be strict. Blow up A2 at 0, andconsider in the y-chart of A2 the transforms of I = (x2, y3). Show that Ig = (x2, y),Is = (x2, 1) = K[x, y].

Example 6.24. Let X = V (f) be a hypersurface in An and Z a regular closedsubvariety which is contained in the locus of points of X where f attains its maximalorder. Let π : An → An be the blowup along Z and let X ′ = V (f ′) be the stricttransform of X. Show that for points a ∈ Z and a′ ∈ E with π(a′) = a theinequality orda′f ′ ≤ ordaf holds.

Example 6.25. Find an example of a variety X for which the dimension of thesingular locus increases under blowup of a closed regular center Z that is containedin the singular locus of X [Hau98].

7. Lecture VII: Resolution Statements

Definition 7.1. A non-embedded resolution of a variety X is a non-singularvariety X together with a proper birational morphism π : X → X which is a regularisomorphism π : X \ E → X \ Sing(X) outside E = π−1(Sing(X)).

Remark 7.2. Requiring properness implies the surjectivity of π and excludestrivial cases as e.g. taking for X the locus of regular points of X and for π theinclusion map. One may ask for additional properties: (a) Any automorphismϕ : X → X of X shall lift to an automorphism ϕ : X → X of X which commuteswith π, say π ◦ ϕ = ϕ ◦π. (b) If X is defined over K and K ⊂ L is a field extension,any resolution of XL should induce a resolution of X = XK.


Definition 7.3. A local non-embedded resolution of a variety X at a pointa is the germ (X, a′) of a non-singular variety X together with a local morphismπ : (X, a′)→ (X, a) inducing an isomorphism of the function fields of X and X.

Definition 7.4. Let X be an affine irreducible variety with coordinate ringR = K[X]. A (local, ring-theoretic) non-embedded resolution of X is a ring exten-sion R ↪→ R of R into a regular ring R having the same quotient field as R.

Definition 7.5. Let X be a subvariety of a regular ambient variety W . Anembedded resolution of X consists of a proper birational morphism π : W → W

from a regular variety W onto W which is an isomorphism over W \ Sing(X) suchthat the total transform X∗ = π−1(X) of X is a subvariety of W with normalcrossings.

Definition 7.6. A strong resolution of a variety X is, for each closed embed-ding of X into a regular variety W , a birational proper morphism π : W → Wsatisfying the following five properties [EH02]:

Embeddedness. The variety W is regular and the total transform X∗ = π−1(X)of X in W has simple normal crossings.

Equivariance. Let W ′ → W be a smooth morphism and let X ′ be the inverseimage of X in W ′. The morphism π′ : W ′ →W ′ induced by π : X →W by takingfiber product of X and W ′ over W is an embedded resolution of X ′. This impliesthe economy of the resolution and also that π commutes with open immersions,localization, completion, automorphisms of W stabilizing X and taking cartesianproducts with regular varieties.

Excision. The restriction τ : X → X of π to X does not depend on the choiceof the embedding of X in W .

Explicitness. The morphism π is a composition of blowups along regular centerstransversal to the exceptional loci of the previous sequence blowups.

Effectiveness. There exists, for all varieties X, a local upper semicontinuousinvariant inva(X) in a well-ordered set Γ which attains its minimal value if andonly if X is regular (or a normal crossings variety) and such that the top locus Sof inva(X) is closed and regular in X and blowing up X along S makes inva(X)drop all points a′ above a ∈ S.

Remark 7.7. One may require in addition that the centers of a resolution aretransversal to the inverse images of a given normal crossings divisor E in W . Thisis known as the boundary condition.

Definition 7.8. Let I be an ideal on a regular ambient variety W . A log-resolution of I is a proper birational morphism π : W →W from a regular varietyW onto W which is an isomorphism over W \ Sing(I) such that I∗ = π−1(I) is alocally monomial ideal on W .

Example 7.9. The blowup of the cusp X : x2 = y3 in A2 at 0 produces a non-embedded resolution. Further blouwps are necessary to make it to an embeddedresolution.

32 HERWIG HAUSER

Example 7.10. Determine the geometry at 0 of the hypersurfaces in A4 definedby the following equations:

(a) x+ x7 − 3yw + y7z2 + 17yzw5 = 0,(b) x+ x5y2 − 3yw + y7z2 + 17yzw5 = 0,(c) x3y2 + x5y2 − 3yw + y7z2 + 17yzw5 = 0.For the first two equation, the implicit function theorem shows that the zerosets

are regular at 0 and gives a formal parametrization. The zeroset of the thirdequation is singular at 0, and one cannot use the theorem to describe it at 0.

Example 7.11. Let Z be the variety in A3C given by the equation (x2− y3)2 =

(z2 − y2)3. Show that the map α : A3 → A3 : (x, y, z) → (u2z3, uyz2, uz2), withu(x, y) = x(y2 − 1) + y, resolves the singularities of Z. What is the inverse imageα−1(Z)? Produce instructive pictures over R.

Example 7.12. Consider the inverse images of the cusp X = V (y2−x3) in A2

under the maps πx, πy : A2 → A2, πx(x, y) = (x, xy), πy(x, y) = (xy, y). Factor themaximal power of x, respectively y, from the equation of the inverse image of Xand show that the resulting equation defines in both cases a regular variety. Applythe same process to the variety E8 = V (x2 + y3 + z5) ⊂ A3 repeatedly until allresulting equations define non-singular varieties.

Example 7.13. Let R be the coordinate ring of an irreducible plane algebraiccurve X. The integral closure R of R in the field of fractions of R is a regular ringand thus resolves R. The resulting curve X is the normalization of X [Mum99]III.8, [dJP00] 4.4.

Example 7.14. Consider a cartesian product X = Y × Z with Z a regularvariety. Show that a resolution of X can be obtained from a resolution Y ′ of Y bytaking the cartesian product Y ′ × Z of Y ′ with Z.

Example 7.15. Let X and Y be two varieties (schemes, analytic spaces) withsingular loci Sing(X) and Sing(Y ) respectively. Suppose that X ′ and Y ′ are regularvarieties (schemes, analytic spaces, within the same category as X and Y ) togetherwith proper birational morphisms π : X ′ → X and τ : Y ′ → Y which defineresolutions of X and Y respectively, and which are isomorphisms outside Sing(X)and Sing(Y ). There is a uniquely determined proper birational morphism f :X ′ × Y ′ → X × Y . So X ′ × Y ′ gives rise to a resolution of X × Y .

8. Lecture VIII: Invariants of Singularities

Definition 8.1. A stratification of an algebraic variety X is a decompositionof X into finitely many disjoint locally closed subvarieties Xi, called the strata,

X =⋃

iXi,

such that the boundaries Xi \Xi of strata are unions of strata. This last propertyis called the frontier condition. Two strata are called adjacent if one lies in theclosure of the other.

Definition 8.2. A local invariant on an algebraic variety X is a functioninv(X) : X → Γ from X to a well-ordered set (Γ,≤) which associates to each pointa ∈ X an element inva(X) depending only on the formal isomorphism class of X at


a: If (X, a) and (X, b) are formally isomorphic, viz OX,a ' OX,b, then inva(X) =invb(X). Usually, the ordering on Γ will also be total: for any c, d ∈ Γ either c ≤ dor d ≤ c holds. The invariant is upper semicontinuous along a subvariety S of X iffor all c ∈ Γ, the sets

topS(inv, c) = {a ∈ S, inva(X) ≥ c}

are closed in S. If S = X, the map inv(X) is called upper semicontinuous.

Remark 8.3. The upper semicontinuity signifies that the value of inv(X) canonly go up or remain the same when passing to a limit point. In the case of schemes,the value of inv(X) has also to be taken into account at non-closed points of X.

Definition 8.4. Let X be a subvariety of a not necessarily regular ambientvariety W defined by an ideal I, and let Z be an irreducible subvariety of W definedby the prime ideal J . The order of X or I in W along Z or J is the maximal integerk = ordZ(X) = ordZ(I) such that IZ ∈ JkZ , where IZ = I ·OW,Z and JZ = J ·OW,Zdenote the ideals generated by I and J in the localization OW,Z of W along Z. IfZ = {a} is a point of W , the order of X and I at a is denoted by orda(X) = orda(I)or ordma

(X) = ordma(I).

Definition 8.5. Let R be a local ring with maximal ideal m. Let k ∈ N bean integer. The k-th symbolic power J (k) of a prime ideal J is defined as the idealgenerated by all elements x ∈ R for which there is an element y ∈ R \ J such thaty · xk ∈ Jk. Equivalently, J (k) = Jk ·RJ ∩R.

Remark 8.6. The symbolic power is the smallest J-primary ideal containingJk. If J is a complete intersection, the ordinary power Jk and the symbolic powerJ (k) coincide [ZS75] IV, §12, [Pel88].

Proposition 8.7. Let X be a subvariety of a not necessarily regular ambientvariety W defined by an ideal I, and let Z be an irreducible subvariety of W definedby the prime ideal J . The order of X along Z is the maximal integer k such thatI ⊂ J (k).

Proof. This follows from the equality Jk ·RJ = (J ·RJ)k. �

Proposition 8.8. Let R be a noetherian local ring with maximal ideal m, andlet R denote its completion with maximal ideal m = m · R. Let I be an ideal of R,with completion I = I · R. Then ordm(I) = ord bm(I).

Proof. If I ⊂ mk, then also I ⊂ mk, and hence ordm(I) ≤ ord bm(I). Con-versely, m = m ∩ R and

⋂i≥0(I + mi) = I ∩ R by Lem. 2.25. Hence, if I ⊂ mk,

then I ⊂ I ∩R ⊂ mk ∩R = mk, so that ordm(I) ≥ ord bm(I). �

Definition 8.9. Let X be a subvariety of a regular variety W , and let a be apoint of W . The local top locus topa(X) of X at a with respect to the order is thestratum S of points of an open neighborhood U of a in W where the order of Xequals the order of X at a. The top locus top(X) of X with respect to the orderis the (global) stratum S of points of W where the order of X attains its maximalvalue. For c ∈ N, define topa(X, c) and top(X, c) as the local and global stratumof points of W where the order of X is at least c.

34 HERWIG HAUSER

Remark 8.10. The analogous definition holds for ideals on W and can be madefor other local invariants. By the upper semicontinuity of the order, the local toplocus of X at a is locally closed in W , and the top locus of X is closed in W .

Proposition 8.11. The order of a variety X or an ideal I of a regular varietyW at points of W defines an upper semicontinuous local invariant on W .

Proof. In characteristic zero, the assertion follows from the next proposition.For the case of positive characteristic, see [Hir64] III.3, Cor. 1, p. 220. �

Proposition 8.12. Over fields of zero characteristic, the local top locus topa(I)of an ideal I at a is defined by the vanishing of all partial derivatives of elementsof I up to order o− 1, where o is the order of I at a.

Proof. In zero characteristic, a polynomial has order o at a point a if andonly if all its partial derivatives up to order o− 1 vanish at a. �

Proposition 8.13. Let X be a subvariety of a regular ambient variety W . LetZ be a non-singular subvariety of W and a a point on Z such that locally at a theorder of X is constant along Z, say equal to d = ordaX = ordZX. Consider theblowup π : W ′ →W of W along Z with exceptional divisor E = π−1(Z). Let a′ bea point on E mapping under π to a. Denote by X∗, Xg and Xs the total, weakand strict transform of X, respectively. Then, locally at a′, the order of X∗ alongE is d, and

orda′Xs ≤ orda′Xg ≤ d.

Proof. By Prop. 5.4 there exist local coordinates x1, . . . , xn of W at a suchthat Z is defined locally at a by x1 = . . . = xk = 0 for some k ≤ n, and such thata′ is the origin of the x1-chart of the blowup. Let I ⊂ OW,a be the local ideal ofX in W , and let f be an element of I. It has an expansion f =

∑α∈Nn cαx

α withrespect to the coordinates x1, . . . , xn, with coefficients cα ∈ K.

Set α+ = (α1, . . . , αk). Since the order of X is constant along Z, the inequality|α+| = α1 + . . . + αk ≥ d holds whenever cα 6= 0. There is an element f with anexponent α such that cα 6= 0 and such that |α| = |α+| = d. The total transformX∗ and the weak transform Xg are given locally at a′ by the ideal generated byall

f∗ =∑α∈Nn

cαx|α+|1 xα2

2 . . . xαnn ,

respectivelyfg =

∑α∈Nn

cαx|α+|−d1 xα2

2 . . . xαnn ,

for f varying on I. The exceptional divisor E is given locally at a′ by the equationx1 = 0. This implies that, locally at a′, ordEf∗ ≥ d for all f ∈ I and ordE f∗ = dfor the special f chosen above with |α| = |α+| = d. Therefore ordEX∗ = d locallyat a′.

Since orda′ fg ≤ |α−| ≤ d for the chosen f , it follows that orda′Xg ≤ d. Theideal Is of the strict transform Xs contains the ideal Ig of the weak transform Xg,thus also orda′Xs ≤ orda′Xg. �

Corollary 8.14. If the order of X is globally constant along Z, the order ofX∗ along E is globally equal to d.


Definition 8.15. Let X be a subvariety of a regular variety W , and let W ′ →W be a blowup with center Z. Denote by X ′ the strict or weak transform in W ′.A point a′ ∈ W ′ above a ∈ Z is called infinitesimally near to a or equiconstant iforda′X ′ = ordaX.

Definition 8.16. Let X be a variety defined over a field K, and let a be apoint of W . The Hilbert-Samuel function HSa(X) : N→ N of X at a is defined by

HSa(X)(k) = dimK(mkX,a/mk+1

X,a ),

where mX,a denotes the maximal ideal of the local ring OX,a of X at a. If X isa subvariety of a regular variety W defined by an ideal I, with local ring OX,a =OW,a/I, one also writes HSa(I) for HSa(X).

Remark 8.17. The Hilbert-Samuel function does not depend on the embeddingof X in W . There exists a univariate polynomial P (t) ∈ Q[t], called the Hilbert-Samuel polynomial of X at a, such that HSa(X)(k) = P (k) for all sufficiently largek [Ser00].

Theorem 8.18. The Hilbert-Samuel function of a subvariety X of a regularvariety W defines an upper semicontinuous local invariant on X with respect to thelexicographic ordering of integer sequences.

Proof. [Ben70] Thm. 4, p. 82. �

Theorem 8.19. Let π : W ′ −→ W be the blowup of W along a non-singularcenter Z. Let I be an ideal of OW . Assume that the Hilbert-Samuel function of Iis constant along Z, and denote by Is the strict transform of I in W ′. Let a′ ∈ Ebe a point in the exceptional divisor mapping under π to a. Then

HSa′(Is) ≤ HSa(I)

holds with respect to the lexicographic ordering of integer sequences.

Proof. [Ben70] Thm. 0. �

Theorem 8.20. (Zariski-Nagata, [Hir64] III.3, Thm. 1, p. 218) Let S ⊂ T beclosed irreducible subvarieties of a closed subvariety X of a regular ambient varietyW . The order of X along T is less than or equal to the order of X along S.

Remark 8.21. In the case of schemes, the assertion says that if X is embeddedin a regular ambient scheme W , and a, b are points of X such that a lies in in theclosure of b, then ordbX ≤ ordaX.

Proof. The proof goes in several steps and relies on the non-embedded res-olution of curves. Let R be a regular local ring, m its maximal ideal, p a primeideal in R and I 6= 0 a non-zero ideal in R. Denote by νp(I) the maximal integerν ≥ 0 such that I ⊂ pν . Recall that ordp(I) is the maximal integer n such thatIRp ⊂ pnRp or, equivalently, I ⊂ p(ν), where p(ν) = pνRp ∩ R denotes the ν-thsymbolic power of p. Thus νp(I) ≤ ordp(I) and the inequality can be strict if p isnot a complete intersection, in which case the usual and the symbolic powers of pmay differ. Write ν(I) = νm(I) for the maximal ideal m so that νp(I) ≤ ν(I) andνp(I) ≤ ν(IRp) = ordp(I).

The assertion of the theorem is equivalent to showing that ν(IRp) ≤ ν(I) takingfor R the local ring OW,S of W along S, for I the ideal of R defining X in W alongS and for p the ideal defining T in W along S.

36 HERWIG HAUSER

If R/p is regular, then νp(I) = ν(IRp) because pnRp ∩ R = pn: The inclusionpn ⊆ pnRp ∩ R is clear, so suppose that x ∈ pnRp ∩ R. Choose y ∈ pn and s 6∈ psuch that xs = y. Then n ≤ νp(y) = νp(xs) = νp(x) + νp(s) = νp(x), hence x ∈ pn.It follows that ν(IRp) ≤ ν(I).

It remains to prove the inequality in the case that R/p is not regular. SinceR is regular, there is a chain of prime ideals p = p0 ⊂ . . . ⊂ pk = m in R withdim(Rpi

/pi−1Rpi) = 1 for 1 ≤ i ≤ n. By induction on the dimension of R/p, it

therefore suffices to prove the inequality in the case dim(R/p) = 1. Since the orderremains constant under completion of a local ring by Prop. 8.8, it can be assumedthat R is complete.

By the embedded resolution of curve singularities there exists a sequence ofcomplete regular local rings R = R0 → R1 → . . . → Rk with prime ideals p0 = pand pi ⊂ Ri with the following properties:

• Ri+1 is a monoidal transform of Ri with center the maximal ideal mi ofRi.

• (Ri+1)pi+1 = (Ri)pi.

• Rk/pk is regular.Set I0 = I and let Ii+1 be the weak transform of Ii under Ri → Ri+1. SetR′ = Rk, I ′ = Ik and p′ = pk. The second condition on the blowups Ri impliesν(IRp) = ν(I ′R′p′). By Prop. 8.13 one knows that ν(I ′) ≤ ν(I). Further, sinceR′/p′

is regular, ν(I ′R′p′) ≤ ν(I ′). Combining these inequalities yields ν(IRp) ≤ ν(I). �

Proposition 8.22. An upper semicontinuous local invariant inv(X) : X →Γ with values in a well and totally ordered set Γ induces, up to refinement, astratification of X with strata Xc = {a ∈ X, inva(X) = c}, for c ∈ Γ.

Proof. For a given value c ∈ Γ, let S = {a ∈ X, inva(X) ≥ c} and T = {a ∈X, inva(X) = c}. The set S is a closed subset of X. If c is a maximal value ofthe invariant on X, the set T equals S and is thus a closed stratum. If c is notmaximal, let c′ > c be an element of Γ which is minimal with c′ > c. Such elementsexist since Γ is well-ordered. As Γ is totally ordered, c′ is unique. Therefore

S′ = {a ∈ X, inva(X) ≥ c′} = {a ∈ X, inva(X) > c} = S \ T

is closed in X. Therefore T is open in S and hence locally closed in X. As Sis closed in X and T ⊂ S, the closure T is contained in S. It follows that theboundary T \ T is contained in S′ and closed in X. Refine the strata by replacingS′ by T \ T and S′ \ T to get a stratification. �

Example 8.23. The singular locus S = Sing(X) of a variety is closed andproperly contained in X. Let X1 = Reg(X) be the set of regular points of X.It is open and dense in X. Repeat the procedure with X \ X1 = Sing(X). Bynoetherianity, the process eventually stops, yielding a stratification of X in regularstrata. The strata of locally minimal dimension are closed and non-singular. Thefrontier condition holds, since the regular points of a variety are dense in the variety,and hence X1 = X = X1 ∪Sing(X).

Example 8.24. There exists a threefold X whose singular locus Sing(X) con-sists of two components, a non-singular surface and a singular curve meeting thesurface at a singular point of the curve. The stratification by iterated singular locisatisfies the frontier condition.


Example 8.25. Give an example of a variety whose stratification by the iter-ated singular loci has four different types of strata.

Example 8.26. Find interesting stratifications for three three-folds.

Example 8.27. The stratification of an upper semicontinuous invariant neednot be finite. Take for Γ the set X underlying a variety X with the trivial (partial)ordering a ≤ b if and only if a = b.

Example 8.28. Let X be the non-reduced scheme defined by xy2 = 0 in A2.The order of X at points of the y-axis outside 0 is 1, at points of the x-axis outside0 it is 2, and at the origin it is 3. The local embedding dimension equals 1 at pointsof the y-axis outside 0, and 2 at all points of the x-axis.

Example 8.29. Determine the stratification given by the order for the fol-lowing varieties. If the smallest stratum is regular, blow it up and determine thestratification of the strict transform. Produce pictures and describe the geometricchanges.

(a) Cross: xyz = 0,(b) Whitney umbrella: x2 − yz2 = 0,(c) Kolibri: x3 + x2z2 − y2 = 0,(d) Xano: x4 + z3 − yz2 = 0,(e) Cusp & Plane: (y2 − x3)z = 0.

Example 8.30. Consider the order ordZI of an ideal I in K[x1, . . . , xn] along aclosed subvariety Z of An, defined as the order of I in the localization of K[x1, . . . , xn]Jof K[x1, . . . , xn] at the ideal J defining Z in An. Express this in terms of the sym-bolic powers J (k) = Jk · K[x1, . . . , xn]J ∩ K[x1, . . . , xn] of J . Give an example toshow that ordZI need not coincide with the maximal power k such that I ⊂ Jk. IfJ defines a complete intersection, the ordinary powers Jk and the symbolic powersJ (k) coincide. This holds in particular when Z is a coordinate subspace of An [ZS75]IV, §12, [Pel88].

Example 8.31. 2 The ideal I = (y2 − xz, yz − x3, z2 − x2y) of K[x, y, z] hassymbolic square I(2) strictly containing I2.

Example 8.32. The variety defined by I = (y2 − xz, yz − x3, z2 − x2y) in A3

has parametrization t 7→ (t3, t4, t5). Let f = x5 +xy3 +z3−3x2yz. For all maximalideals m which contain I, f ∈ m2 and thus, ordmf ≥ 2. But f 6∈ I2. Since xf ∈ I2

and x does not belong to I, it follows that f ∈ I2RI and thus ordIf ≥ 2, in fact,ordIf = 2.

Example 8.33. The order of an ideal depends on the embedding of X in W .If X is not minimally embedded locally at a, the order of X at a is 1 and notsignificant.

Example 8.34. Associate to a stratification of a variety the so called Hassediagram, i.e., the directed graph whose nodes and edges correspond to strata, re-spectively to the adjacency of strata. Determine the Hasse diagram for the surfaceX given in A4 as the cartesian product of the cusp C : x2 = y3 with the node D :x2 = y2+y3. Then project X to A3 by means of A4 → A3, (x, y, z, w) 7→ (x, y+z, w)and compute the Hasse diagram of the image Y of X under this projection.

2This example, due to Macaulay, was kindly communicated by M. Hochster.

38 HERWIG HAUSER

Example 8.35. Show that the order of a hypersurface, the dimension andthe Hilbert-Samuel function of a variety, the embedding-dimension of a varietyand the local number of irreducible components are invariant under local formalisomorphisms, and determine whether they are upper or lower semicontinuous. Howdoes each of these invariants behave under localization and completion?

Example 8.36. Take inva(X) = dima(X), the dimension of X at a. It is con-stant on irreducible varieties, and upper semicontinuous on arbitrary ones, becauseat an intersection point of several components, dima(X) is the maximum of thedimensions of the components.

Example 8.37. Take inva(X) = the number of irreducible components of Xpassing through a. If the components carry multiplicities as e.g. a divisor, one mayalternatively take the sum of the multiplicities of the components passing througha. Both options produce an upper semicontinuous local invariant.

Example 8.38. Take inva(X) = embdima(X) = dim TaX, the local embed-ding dimension of X at a. It is upper semicontinuous. At regular points, it equalsthe dimension of X at a, at singular points it exceeds this dimension.

Example 8.39. Consider for a given coordinate system x, y1, . . . , yn on An apolynomial of order c at 0 of the form

g(x, y1, . . . , yn) = xc +c−1∑i=0

gi(y) · xi.

Express the order and the top locus of g nearby 0 in terms of the orders of thecoefficients gi.

Example 8.40. Take inva(X) = HSa(X), the Hilbert-Samuel function of Xat a. The lexicographic order on integer sequences defines a well-ordering on Γ ={γ : N→ N}. Find two varieties X and Y with points a and b where HSa(X) andHSb(Y ) only differ from the fourth entry on.

Example 8.41. Take inva(X) = ν∗a(X) the increasingly ordered sequence ofthe orders of a minimal Macaulay basis of the ideal I defining X in W at a [Hir64]III.1, def. 1 and Lem. 1, p. 205. It is upper semicontinuous but does not behavewell under specialization [Hir64] III.3, Thm. 2, p. 220 and the remark after Cor. 2,p. 220, see also [Hau98], ex. 12.

Example 8.42. Let be given a monomial order <ε on Nn, i.e., a total orderingwith minimal element 0 which is compatible with addition in Nn. The initial idealin(I) of an ideal I of K[[x1, . . . , xn]] with respect to <ε is the ideal generated by allinitial monomials of elements f of I, i.e., the monomials with minimal exponentwith respect to <ε in the series expansion of f . The initial monomial of 0 is 0.

The initial ideal is a monomial ideal in K[[x1, . . . , xn]] and depends on thechoice of coordinates. If the monomial order <ε is compatible with degree, in(I)determines the Hilbert-Samuel function HSa(I) of I [Hau04].

Order monomial ideals totally by comparing their increasingly ordered uniqueminimal monomial generator system lexicographically, where any two monomialgenerators are compared with respect to <ε. If two monomial ideals have generatorsystems of different length, complete the sequences of their exponents by a symbol


∞ so as to be able to compare them. This defines a well-order on the set ofmonomial ideals.

Take for inva(X) the minimum min(I) or the maximum max(I) of the initialideal of the ideal I of X at a, the minimum and maximum being taken over allchoices of local coordinates, say regular parameter systems of K[[x1, . . . , xn]]. Bothexist and define local invariants which are upper semicontinuous with respect tolocalization [Hau04] Thms. 3 and 8.

(a) The minimal initial ideal min(I) = minx{in(I)} over all choices of regularparameter systems of K[[x1, . . . , xn]] is achieved for almost all regular parametersystems.

(b) * Let I be an ideal in K[x1, . . . , xn] and denote by Ia the induced ideal inK[[x1−an, . . . , xn−an]]. The minimal initial ideal min(Ia) is upper semicontinuouswhen the point a varies.

(c) Compare the induced stratification of An with the stratification by theHilbert-Samuel function of I.

(d) Let Z be a regular center inside a stratum of the in(I) stratification, andconsider the induced blowup An → An along Z. Let a ∈ Z and a′ ∈W ′ be a pointabove a. Assume that the monomial order <ε is compatible with degree. Showthat mina(I) and maxa(I) do not increase when passing to the strict transform ofIa at a′ [Hau04] Thm. 6.

Example 8.43. Let (W ′, a′) → (W,a) be the composition of two monomialpoint blowups of W = A2 with respect to coordinates y, z, defined as follows. Thefirst is the blowup of A2 with center 0 considered at the origin of the y-chart, thesecond has as center the origin of the y-chart and is considered at the origin of thez-chart. Show that the order of the weak transform g′(y, z) at a′ of any non zeropolynomial g(y, z) in W is at most the half of the order of g(y, z) at a.

Example 8.44. * Let (W ′, a′) → (W,a) be a composition of local blowups inregular centers such that a′ lies in the intersection of n exceptional componentswhere n is the dimension of W at a. Let f ∈ OW,a and assume that the character-istic is zero. Show that the order of f has dropped between a and a′.

Example 8.45. * Show the same in positive characteristic.

Example 8.46. (B. Schober) Let K be a non perfect field of characteristic 3,let t ∈ K \ K2 be an element which is not a square. Stratify the hypersurfaceX : x2 + y(z2 + tw2) = 0 in A4 according to its singularities. Show that thisstratification does not provide suitable centers for a resolution.

Example 8.47. Let I = (x2 + y17) be the ideal defining an affine plane curvesingularity X with singular locus the origin of W = A2. The order of X at 0 is 2.The blowup π : W ′ → W of W at the origin with exceptional divisor E is coveredby two affine charts, the x- and the y-chart. The total and strict transform of I inthe x-chart are as follows:

I∗ = (x2 + x17y17),Is = (1 + x15y17).

At the origin of this chart, the order of Is has dropped to zero, so the stricttransform Xs of X does not contain this point. Therefore it suffices to consider

40 HERWIG HAUSER

the complement of this point in E, which lies entirely in the y-chart. There, oneobtains the following transforms:

I∗ = (x2y2 + y17),Is = (x2 + y15).

The origin a′ of the y-chart is the only singular point of Xs. The order of Xs at a′

has remained constant equal to 2. Find a local invariant of X which has improvedat a′. Make sure that it does not depend on any choices.

Example 8.48. The ideal I = (x2 +y16) has in the y-chart the strict transformIs = (x2 + y14). If the ground field has characteristic 2, the y-exponents 16 and 14are irrelevant because the coordinate change x 7→ x+ y8 transforms I into (x2).

Example 8.49. The ideal I = (x2 + 2xy7 + y14 + y17) has in the y-chart thestrict transform Is = (x2 + 2xy6 + y12 + y15). Here the drop of the y-exponent ofthe last monomial from 17 to 15 is significant, whereas the terms 2xy7 + y14 canbe eliminated in any characteristic by the coordinate change x 7→ x+ y7.

Example 8.50. The ideal I = (x2 +xy9) = (x)(x+y9) defines the union of twonon-singular curves in A2 which have a common tangent line at their intersectionpoint 0. The strict transform is Is = (x2 + xy8). The degree of tangency, viz theintersection multiplicity, has decreased.

Example 8.51. Let X and Y be two non-singular curves in A2, meeting at onepoint a. Show that there exists a sequence of point blowups which separates thetwo curves, i.e., so that the strict transforms of X and Y do not intersect.

Example 8.52. Take I = (x2 + g(y)) where g is a polynomial in y with orderat least 3 at 0. The strict transform in the y-chart is Is = x2 + y−2g(y), with orderat 0 equal to 2 again. This suggests to take the order of g as a secondary invariant.In characteristic 2 it may depend on the choice of coordinates.

Example 8.53. Take I = (x2 + xg(y) + h(y)) where g and h are polynomialsin y of order at least 1, respectively 2, at 0. The order of I at 0 is 2. The stricttransform equals Is = (x2 + xy−1g(y) + y−2h(y)) of order 2 at 0. Here it is lessclear how to detect a secondary invariant which represents an improvement.

Example 8.54. Take I = (x2 + y3z3) in A3, and apply the blowup of A3 inthe origin. The strict transform of I in the y-chart equals Is = (x2 + y4z3) and thesingularity seems to have got worse.

Example 8.55. Let X be a surface in three-space, and S its top locus. AssumeS is singular at a, and let X ′ be the blowup of X in a. Determine the top locus ofX ′ [Zar44, Hau00].

9. Lecture IX. Hypersurfaces of Maximal Contact

Proposition 9.1. (Zariski) Let X be a subvariety of a regular variety W ,defined over a field of arbitrary characteristic. Let W ′ → W be the blowup of Walong a regular center Z contained in the top locus of X, and let a be a point ofZ. There exists, in a neighbourhood U of a, a regular closed hypersurface V of Uwhose strict transform V s in W ′ contains all points a′ of W ′ lying above a wherethe order of the strict transform Xs of X has remained constant.


Proof. Choose local coordinates x1, . . . , xn of W at a. The associated gradedring of OW,a can be identified with K[x1, . . . , xn]. Let in(I) ⊂ K[x1, . . . , xn] denotethe ideal of inital forms of elements of I at a. Apply a linear coordinate changeso that generators of in(I) are expressed with the minimal number of variables,say x1, . . . , xk, for some k ≤ n. Choose any 1 ≤ i ≤ k and define V in W at a byxi = 0. It follows that the local top locus of X at a is contained in V . Hence Z ⊂ V ,locally at a. Let a′ be a point of W ′ above a where the order of X has remainedconstant. By Prop. 5.4 the local blowup (W ′, a′)→ (W,a) can be made monomialby a suitable coordinate change. The assertion then follows by computation, cf. ex.9.8 and [Zar44]. �

Definition 9.2. Let X be a subvariety of a regular variety W , and let a bea point of W . A hypersurface of maximal contact for X at a is a regular closedhypersurface V of an open neighborhood U of a in W such that

(1) V contains the local top locus S of X at a, i.e., the points of U where theorder of X equals the order of X at a.

(2) The strict transform V s of V under any blowup of U along a regular centerZ contained in S contains all points a′ above a where the order of Xs has remainedconstant equal to the order of X at a, i.e., V is a directrix of X at a.

(3) Property (2) is preserved in any sequence of blowups with regular centerscontained in the successive top loci of the strict transforms of X along which theorder of X has remained constant.

a′ ∈ E ∩ V s ⊂W ′

↓ ↓ ↓ π

a ∈ Z ⊂ V ⊂W

Definition 9.3. Assume that the characteristic of the ground field is zero.Let X be a subvariety of a regular variety W defined locally at a point a of W bythe ideal I. Let o be the order of X at a. An osculating hypersurface for X at ais a regular closed hypersurface V of a neighbourhood U of a in W defined by aderivative of order o− 1 of an element f of I [EH02].

Remark 9.4. The element f has necessarily order o at a, and its (o− 1)-st de-rivative has order 1 at a, so that it defines a regular hypersurface at a. The conceptis due to Abhyankar and Zariski [AZ55]. Abhyankar called the local isomorphismconstructing an osculating hypersurface V from a given regular hypersurface H ofW Tschirnhaus transformation. If H is given by xn = 0, this transformationeliminates from f all monomials whose xn-exponent is o − 1. The concept wasexploited by Hironaka in his proof of characteristic zero resolution [Hir64].

For each point a in X, osculating hypersurfaces contain locally at a the localtop locus S = topa(X) and the equiconstant points above.

Proposition 9.5. (Abhyankar, Hironaka) Let X be a subvariety of a regularvariety W , and let a be a point of W . For ground fields of characteristic zero thereexist, locally at a in W , hypersurfaces of maximal contact for X. Any osculatinghypersurface V at a has maximal contact with X at a.

Proof. [EH02]. �

42 HERWIG HAUSER

Remark 9.6. The assertion of the proposition does not hold over fields ofpositive characteristic: Narasimhan gave an example of a hypersurface X in A4

over a field of characteristic 2 whose top locus is not contained at 0 in any regularlocal hypersurface, and Kawanoue describes a whole family of such varieties [Nar83,Kaw13], [Hau98], ex. 8. See also ex. 12.1 below. In Narasimhan’s example, there isa sequence of point blowups for which there is no regular local hypersurface V ofA4 at 0 whose strict transforms contain all points where the transforms of X haveorder 2 as at the beginning [Hau03] II.14, ex. 2, p. 388.

Remark 9.7. The existence of hypersurfaces of maximal contact in zero char-acteristic suggests to associate to X locally at a point a a variety Y or an idealJ in V whose transforms under the blowup of V and W along a center Z locallycontained in V equals the variety Y ′ associated to the strict transform Xs of X inV s at points a′ above a where the order of Xs has remained constant. The varietyY or the ideal J and their transforms may then help to measure the improvementof Xs at a′ by looking at their respective orders. This is precisely the way howthe proof of resolution in zero characteristic proceeds. The reasoning is also knownas descent in dimension. The main problem in this approach is to define properlythe variety Y , respectively the ideal I, and to show that the local construction isindependent of the choice of V and patches to give a global resolution algorithm.

a′ ∈ E ⊂W ′ V ′ ⊃ Y ′

↓ ↓ ↓ π ↓ ↓ π|Y ′

a ∈ Z ⊂W V ⊃ Y

Example 9.8. Let π : (W ′, a′) → (W,a) be a local blowup and let x1, . . . , xnbe local coordinates on W at a such that π is monomial. Assume that x1 appears inthe initial form of an element f ∈ OW,a, and let V ⊂W be the local hypersurface ata defined by x1 = 0. If the order of the strict transform fs of f at a′ has remainedconstant equal to the order of f at a, the point a′ belongs to the strict transformV s of V .

Example 9.9. Let the characteristic of the ground field be different from 3.Apply the second order differential operator ∂ = ∂2

∂x2 to f = x3 + x2yz + z5 sothat ∂f = 6x + 2yz. This defines a hypersurface of maximal contact for f at 0.Replacing in f the variable x by x− 1

3yz gives

g = x3 − 13xy2z2 +

227y3z3 + z5.

The term of degree 2 in x has been eliminated, and x = 0 defines an osculatinghypersurface for g at 0.

Example 9.10. Let X ⊂ An be a hypersurface defined locally at the origin by apolynomial f = xdn+

∑d−1i=0 ai(y)xin where y = (x1, . . . , xn−1) and ord0ai(y) ≥ d− i.

Make the change of coordinates xn 7→ xn − 1dad−1(y). Show that after this change,

the hypersurface defined by xn = 0 has maximal contact with X at the origin.What prevents this technique from working in positive characteristic?

Example 9.11. Consider the hypersurface X ⊂ A3 given by the equationx2y + xy2 − x2z + y2z − xz2 − yz2. Show that the hypersurface V given by x = 0


does not have maximal contact with X at 0. In particular, consider the blowup ofA3 in the origin. Find a point a′ on the exceptional divisor that lies in the x-chartof the blowup such that the order of the strict transform of X has order 3 at a′.Then show that the strict transform of V does not contain this point.

Example 9.12. Hypersurfaces of maximal contact are only defined locally.They need not patch to give a globally defined hypersurface of maximal contact onW . Find an example for this.

Example 9.13. Consider f = x4 + y4 + z6, g = x4 + y4 + z10 and h = xy+ z10

under point blowup. Determine, according to the characteristic of the ground field,the points where the order of f has remained constant.

Example 9.14. Let f = xc + g(y1, . . . , ym) ∈ K[[x, y1, . . . , ym]] be a formalpower series with g a series of order ≥ c at 0. Show that there exists a formalcoordinate change maximizing the order of g.

Example 9.15. Let f = xc + g(y1, . . . , ym) ∈ K[x, y1, . . . , ym] be a polynomialwith g a polynomial of order ≥ c at 0. Does there exist a local coordinate changemaximizing the order of g?

Example 9.16. Let f be a polynomial or power series in n variables x1, . . . , xnof order c at 0. Assume that the ground field is infinite. There exists a linearcoordinate change after which f(0, . . . , 0, xn) has order c at 0. Such polynomialsand series, called xn-regular of order c at 0, appear in the Weierstrass preparationtheorem, which was frequently used by Abhyankar.

10. Lecture X. Coefficient Ideals

Definition 10.1. Let I be an ideal on W , let a be a point of W with openneighbourhood U , and let V be a regular closed hypersurface of U containing a.Let x1, . . . , xn be coordinates on U such that V is defined in U by xn = 0. Therestrictions of x1, . . . , xn−1 to V form coordinates on V and will be abbreviated byx′. For f ∈ OU , denote by

∑af,i(x′) · xin the expansion of f with respect to xn,

with coefficients af,i = af,i(x′) ∈ OV . The coefficient ideal of I in V at a is theideal JV (I) on V defined by

JV (I) =o∑i=0

(af,i, f ∈ I)o!

o−i .

Remark 10.2. The coefficient ideal is defined on whole V . It depends on thechoice of the coordinates x1, . . . , xn on U , even so the notation only refers to V .Actually, JV (I) depends on the choice of a section OV → OU of the restrictionmap OU → OV . The same definition applies to stalks of ideals in W at points a,giving rise to an ideal, also denoted by JV (I), in the local ring OV,a. The weightso!o−i in the exponents are chosen so as to obtain a systematic behaviour of thecoefficient ideal under blowup, cf. Prop. 884 below. The chosen algebraic definitionof the coefficient ideal is modelled so as to commute with blowups [EH02], but isless conceptual than definitions through differential operators proposed and usedby Encinas-Villamayor, Bierstone-Milman, W lodarczyk, Kawanoue-Matsuki andHironaka [EV00, EV98, BM97, W lo05, KM10, Hir03].

44 HERWIG HAUSER

Proposition 10.3. The passage to the coefficient ideal JV (I) of I in V com-mutes with taking germs along the local top locus S = topa(I) of points of V wherethe order of I in W is equal to the order of I in W at a: Let x1, . . . xn be coor-dinates of W at a, defined on an open neighborhood U of a, and let V be closedand regular in U , defined by xn = 0. The stalks of JV (I) at points b of S inside Ucoincide with the coefficient ideals of the stalks of I at b.

Proof. Clear from the definition of coefficient ideals. �

Corollary 10.4. In the above situation, for any fixed closed hypersurface Vin U ⊂ W open, the order of JV (I) at points of S is upper semicontinuous alongS, locally at a.

Remark 10.5. In general, V need not contain, locally at a, the top locus of Iin W . This can, however, be achieved in zero characteristic by choosing for V anosculating hypersurface, cf. Prop. 10.9 below. In this case, the order of JV (I) atpoints of S does not depend on the choice of the hypersurface, cf. Prop. 10.13. Inarbitrary characteristic, a local hypersurface V will be chosen separately at eachpoint b ∈ S in order to maximize the order of JV (I) at b, cf. Prop. 10.11. Inthis case, the order of JV (I) at b does not depend on the choice of V , and itsupper semicontinuity as b moves along S holds again, but is more difficult to prove[Hau04].

Proposition 10.6. The passage to the coefficient ideal JV (I) of I at a com-mutes with blowup: Let π : W ′ → W be the blowup of W along a regular centerZ contained locally at a in S = topa(I). Let V be a local regular hypersurface ofW at a containing Z and such that V s contains all points a′ above a where theorder of the weak transform Ig has remained constant equal to the order of I ata. For any such point a′, the coefficient ideal JV s(Ig) of Ig equals the controlledtransform JV (I)! = I−cE · JV (I)∗ of JV (I) with respect to the control c = o! witho = orda(I).

Proof. Write V ′ for V s, and let h = 0 be a local equation of E ∩ V ′ in V ′.For f ∈ I of order o at a write the strict transform of f (up to multiplication byunits in the local ring) as fs = h−o · f∗, where ∗ denotes the total transform. Thecoefficients af,i of the monomials xα of the expansion of an element f of I of ordero at a in the coordinates x1, . . . , xn satisfy afg,i = hi−o · (af,i)∗. Then

JV ′(Ig) = JV ′(∑i<o′

afs,i · xin, fg ∈ Ig) =

= JV ′(∑i<o′

ah−o·f∗,i · xin, fg ∈ Ig) =

= JV ′(∑i<o

h−o · (af,i · xin)∗, f ∈ I) =

=∑i<o

h−o! · (a∗f,i, f ∈ I)o!/(o−i) =

= h−o! · (∑i<o

(af,i, f ∈ I)o!/(o−i))∗ =

= h−o! · (JV I)∗ = (JV I)!.This proves the claim. �


Remark 10.7. The definitions of the coefficient ideal used in [EV00, EV98,BM97, W lo05, KM10, Hir03] produce a weaker commutativity property with re-spect to blowups, typically only for the radicals of the coefficient ideals.

Remark 10.8. The order of the coefficient ideal JV (I) of I is not directlysuited as a secondary invariant after the order of I since, by the proposition, thecoefficient ideal passes under blowup to its controlled transform, and thus its ordermay increase. It is appropriate to decompose JV (I) and (JV (I))! into products ofideals with an exceptional monomial factor and a second, possibly singular factor,called the residual factor, which passes under blowup to its weak transform. Theorder of this second factor does not increase under blowup by Prop. 8.13 and canthus serve as a secondary invariant.

Proposition 10.9. Assume that the characteristic of the ground field is zero.Let I have maximal order o at a point a ∈W , and let V be a regular hypersurfacefor I at a, with coefficient ideal J = JV (I). The locus topa(J, o!) of points of Vwhere J has order ≥ o! coincides with topa(I),

topa(J, o!) = topa(I).

Proof. Choose local coordinates x1, . . . , xn in W at a so that V is defined byxn = 0. Expand the elements f of I with respect to xn with coefficients af,i ∈ OV,a,and choose representatives of them on a suitable neighbourhood of a. Let b be apoint in a sufficiently small neighborhood of a. Then, by the upper semicontinuityof the order, b belongs to topa(I) if and only if ordbI ≥ o, which is equivalent to∑i<o af,i · xin having order ≥ o at b for all f ∈ I. This, in turn, holds if and only if

af,i has order ≥ o− i at b, say ao!

o−i

f,i has order ≥ o! at b. Hence b ∈ topa(I) if andonly if b ∈ topa(JV (I), o!). �

Corollary 10.10. Assume that the characteristic of the ground field is zero.Let a be a point in W and set S = topa(I). Let U be a neighbourhood of a onwhich there exists a closed regular hypersurface V which is osculating for I at allpoints of S ∩ U . Let JV (I) be the coefficient ideal of I in V .

(a) The top locus topa(JV (I)) of JV (I) on V is contained in topa(I).(b) The blowup of U along a regular locally closed subvariety Z of topa(JV (I))

Z commutes with the passage to the coefficient ideals of I and Ig in V and V s.

Proof. Assertion (a) is immediate from the proposition, and (b) follows fromProp. 10.6. �

Proposition 10.11. (Encinas-Hauser) Assume that the characteristic of theground field is zero. The order of the coefficient ideal JV (I) of I at a with respectto an osculating hypersurface V at a attains the maximal value of the orders of thecoefficient ideals over all local regular hypersurfaces. In particular, it is independentof the choice of V .

Proof. Choose local coordinates x1, . . . , xn in W at a such that the appro-priate derivative of the chosen element f ∈ I is given by xn. Let o be the order off at a. The choice of coordinates implies that the expansion of f with respect toxn has a monomial xon with coefficient 1 and no monomial in xn of degree o − 1.Any other local regular hypersurface U is obtained from V by a local isomorphismϕ of W at a. Assume that the order of JV (ϕ∗(I)) is larger than the order of

46 HERWIG HAUSER

JV (I). Let g = ϕ∗(f). This signifies that the order of all coefficients ao!

o−i

g,i is largerthan the order of JV (I). Therefore ϕ∗ must eliminate from f the terms of af,i for

which ao!

o−i

f,i has order equal to the order of JV (I). But then ϕ∗ produces from xon anon-zero coefficient ag,o−1 such that ao!g,o−1 has order equal to the order of JV (I),contradiction. �

Remark 10.12. This result suggests to look in positive characteristic for localregular hypersurfaces which maximize the order of the associated coefficient ideal,called hypersurfaces of weak maximal contact [EH02]. They are used in recentapproaches to resolution of singularities in positive characteristic [Hir12, Hau10a],based on the work of Moh [Moh87].

Proposition 10.13. Let the characteristic of the ground field be arbitrary.The supremum in N∪{∞} of the orders of the coefficient ideals JV (I) of I in localregular hypersurfaces V in W at a is realized by a formal hypersurface V in W ata. If the supremum is finite, it can be realized by a local regular hypersurface V inW at a.

Proof. If the supremum is finite, the existence of some V realizing this valueis obvious. If the supremum is infinite, one uses the m-adic completeness of OW,ato construct V , see [EH02, Hau04]. �

Definition 10.14. A formal local regular hypersurface V realizing the supre-mum of the order of the coefficient ideal JV (I) of I at a is called a hypersurface ofweak maximal contact of I at a. If the supremum is finite, it will always be assumedto be a local hypersurface.

Proposition 10.15. (Zariski) Let V be a local formal regular hypersurface inW at a of weak maximal contact with I at a. Let π : W ′ →W be the blowup of Walong a closed regular center Z contained locally at a in S = topa(I). The pointsa′ ∈ W ′ above a for which the order of the weak transform Ig of I at a′ has notdecreased are contained in the strict transform V s of V .

Proof. By definition of weak maximal contact, the variable xn defining V inW at a appears in the initial form of some element f of I of order o = topa(I) ata, cf. ex. 10.23 below. The argument then goes analogous to the proof of Prop.9.1. �

Remark 10.16. In characteristic zero and if V has been chosen osculating ata, the hypersurface V ′ is again osculating at points a′ above a where orda′(I ′) =orda(I), hence it has again weak maximal contact with I ′ at such points a′. Inpositive characteristic this is no longer true, cf. ex. 10.24.

Example 10.17. Determine in all characteristics the points of A2 where thestrict transform of g = x4 + kx2y2 + y4 + 3y7 + 5y8 + 7y9 under the blowup of A2

at 0 has order 4, for any k ∈ N.

Example 10.18. Determine for all characteristics the maximal order of thecoefficient ideal of I = (x3 + 5y3 + 3(x2y2 + xy4) + y6 + 7y7 + y9 + y10) in regularlocal hypersurfaces V at 0.

Example 10.19. Same as before for I = (xy + y4 + 3y7 + 5y8 + 7y9).


Example 10.20. Compute the coefficient ideal of I = (x5 + x2y4 + yk) in thehypersurfaces x = 0, respectively y = 0. According to the value of k and thecharacteristic, which hypersurface is osculating or has weak maximal contact?

Example 10.21. Consider f = x2 + y3z3 + y7 + z7. Show that V ⊂ A3 definedby x = 0 is a local hypersurface of weak maximal contact for f . Blow up A3 at theorigin, respectively along a coordinate axis. How does the order of the coefficientideal of f in V behave under these blowups at the points where the order of fhas remained constant? Factorize the controlled transform of the coefficient idealwith respect to the exceptional factor and observe the behaviour of the order of theresidual factor.

Example 10.22. Assume that, for a given coordinate system x1, . . . , xn in W ata, the coefficient ideal of a polynomial f in the hypersurface V defined by xn = 0 isa principal monomial ideal. Show that there is a sequence of blowups in coordinatesubspaces of the induced affine charts which eventually makes the order of f drop.

Example 10.23. The variable xn defining a hypersurface V in W at a of weakmaximal contact with an ideal I appears in the initial form of some element f of Iof order o = orda(I) at a.

Example 10.24. In positive characteristic, a hypersurface of weak maximalcontact for an ideal I need not have again weak maximal contact after blowup withthe weak transform Ig at points a′ where the order of Ig has remained constant.

Example 10.25. Compute in the following situations the coefficient ideal ofI in W at a with respect to the given local coordinates x, y, z in A3 and thehypersurface V . Determine in each case whether the order of the coefficient idealis maximal. If not, find a coordinate change which maximizes it.

(a) a = 0 ∈ A1, x, V : x = 0, I = (x) and I = (x+ x2).(b) a = 0 ∈ A2, x, y, V : x = 0, I = (x), I = (x+ y2), I = (y + x2), I = (xy).(c) a = (1, 0, 0) ∈ A3, x, y, z, V : y + z = 0, I = (x2), I = (xy), I = (x3 + z3).(d) a = 0 ∈ A3, x, y, z, V : x = 0, I = (xyz), I = (x2 + y3 + z5).(e) a = 0 ∈ A2, x, y, V : x = 0, I = (x2 + y4, y4 + x2).

Example 10.26. Blow up in each of the preceding examples the origin anddetermine the points of the exceptional divisor E where the order of the weaktransform Ig of I has remained constant. Check at these points whether commu-tativity holds for the descent to the coefficient ideal and its controlled transform.

Example 10.27. * Show that the maximum of the order of the coefficient idealJV (I) of a principal idela I over all regular parameter systems of OAn,0 is attained(it might be ∞).

Example 10.28. * Show that this maximum, when taken at any point of thetop locus S of An where I has maximal order o, defines an upper semicontinuousfunction on S.

Example 10.29. Let V be the hypersurface xn = 0 of An and let V ′ → V bethe blowup with center Z in V . Let f be a polynomial of order o at a point a ofZ, with strict transform f ′. Assume that orda′f ′ = ordaf at the origin a′ of thexn-chart. Compare the total transform of the coefficent ideal JV (f) of f with itscontrolled transform with respect to the control c = o!.

48 HERWIG HAUSER

Example 10.30. Show that, in characteristic zero, the top locus of an ideal Iof W , when taken locally at a point a, is contained in a local regular hypersurface Vthrough a, and that this hypersurface maximizes the order of the coefficient ideal.

Example 10.31. Show that the supremum of the order of the coefficient idealof an ideal I in W at a point a can be realized, if the supremum is finite, by a regularsystem of parameters of the local ring OW,a without passing to the completion.

Example 10.32. Consider f = x3 + y2z in A3 and the point blowup of A3 atthe origin. Find, according to the characteristic, at all points of the exceptionaldivisor a hypersurface of weak maximal contact for the strict transform of f .

Example 10.33. Consider surfaces of the form f = xo + yazb · g(y, z) whereyazb is considered as an exceptional monomial factor of the coefficient ideal in thehypersurface V defined x = 0 (up to raising the coefficient ideal to the powerc = o!). Assume that a + b + ord0g ≥ p. Give three examples where the order ofg at 0 is not maximal over all choices of local hypersurfaces at 0, and indicate thecoordinate change which makes it maximal.

Example 10.34. Consider surfaces of the form f = xo + yazb · g(y, z) whereyazb is considered again as an exceptional monomial factor of the coefficient idealin the hypersurface V defined x = 0. Assume that a+ b+ ord0g ≥ o. Compute thestrict transform f ′ = xo + ya

′zb′ · g′(y, z) of f under point blowup at points where

the order of f has remained equal to o. Find three examples where the order of g′

is not maximal over all local coordinate choices.

Example 10.35. For a flag of local regular hypersurfaces Vn−1 ⊃ . . . ⊃ V1 ata one gets from I a chain of coefficient ideals Jn−1, . . . , J1 defined recursively byJi = JVi(Ii+1), where Ji+1 is decomposed into Ji+1 = Mi+1 · Ii+1 with exceptionalmonomial factor Mi+1 and residual factor Ii+1. Show that the lexicographic max-imum of the vector of orders of the ideals Ji at a over all choices of flags at a canbe realized stepwise, choosing first a local hypersurface Vn−1 in W at a maximizingthe order of Jn−1 at a, then a local hypersurface Vn−2 in Vn−1 at a maximizing theorder of Jn−2, and iterating this process.

11. Lecture XI: Resolution in Zero Characteristic

The inductive proof of resolution of singularities in characteristic zero requires amore detailed statement about the nature of the resolution:

Theorem 11.1. Let W be a regular ambient variety and let E ⊂ W be adivisor with normal crossings. Assume that the characteristic of the ground fieldis 0. Let J be an ideal on W , together with a decomposition J = M · I into aprincipal monomial ideal M , the monomial factor of J , supported on a normalcrossings divisor D transversal to E, and an ideal I, the residual factor of J . Letc+ ≥ 1 be a given number, the control of J .

There exists a sequence of blowups of W along regular centers Z transversalto E and D and their total transforms, contained in the locus top(J, c+) of pointswhere J and its controlled transforms with respect to c+ have order ≥ c+, andsatisfying the requirements equivariance and excision of a strong resolution so thatthe order of the controlled transform of J with respect to c+ drops eventually atall points below c+.


Definition 11.2. In the situation of the theorem, with prescribed divisor Dand control c+, the ideal J is called resolved with respect to D and c+ if the orderof J at all points of W is < c+.

Remark 11.3. Once the order of the controlled transform of J has droppedbelow c+, induction on the order can be applied to find an additional sequence ofblowups which makes the order of the controlled transform of J drop everywhereto 0. At that stage, the controlled transform of J has become the whole coordinatering of W , and the total transform of J , which differs from the controlled transformby a monomial exceptional factor, has become a monomial ideal supported on anormal crossings divisor D transversal to E. This establishes the existence of astrong embedded resolution of J , respectively of the singular variety X defined byJ in W .

Proof. The technical details can be found in [EH02], and motivations aregiven in [Hau03]. The main argument is the following.

The resolution process has two different stages: In the first, a sequence ofblowups is chosen through a local analysis of the singularities of J and by inductionon the ambient dimension. The order of the weak transforms of I will be forcedto drop eventually below the maximum of the order of I at the points a of W .By induction on the order one can then apply additional blowups until the orderof the weak transform of I has become equal to 0. At that moment, the weaktransform of I equals the coordinate ring/structure sheaf of the ambient variety,and the controlled transform of J has become a principal monomial ideal supportedon a suitable transform of D, which will again be a normal crossings divisor meetingthe respective transform of E transversally.

For simplicity, denote this controlled transform again by J . It is a principalmonomial ideal supported by exceptional components. The second stage of theresolution process makes the order of J drop below c+. The sequence of blowupsis now chosen globally according to the multiplicities of the exceptional factorsappearing in J . This is a completely combinatorial and quite simple procedurewhich can be found in many places in the literature [Hir64, ?, EH02].

The first part of the resolution process is much more involved and will bedescribed now. The centers of blowup are defined locally at the points where theorder of I is maximal. Then it is shown that the definition does not depend on thelocal choices and thus defines a global, regular and closed center in W . Blowing upW along this center will improve the singularities of I, and finitely many furtherblowups will make the order of the weak transform of I drop.

The local definition of the center goes as follows. Let S = top(I) be the stratumof points in W where I has maximal order. Let a be a point of S and denote byo the order of I at a. Choose an osculating hypersurface V for I at a in an openneighbourhood U of a in W . This is a closed regular hypersurface of U given bya suitable partial derivative of an element of I of order o− 1. The hypersurface Vmaximizes the order of the coefficient ideal Jn−1 = JV (I) over all choices of localregular hypersurfaces, cf. Prop. 10.11. Thus orda(Jn−1) does not depend on thechoice of V . There exists a closed stratum T in S at a of points b ∈ S where theorder of Jn−1 equals the order of Jn−1 at a.

The ideal Jn−1 on V together with the new control c = o! can now be resolved byinduction on the dimension of the ambient space, applying the respective statementof the theorem. The new normal crossings divisor En−1 in V is defined as V ∩ E.

50 HERWIG HAUSER

For this it is necessary that the hypersurface V can be chosen transversal to E. Thisis indeed possible, but will not be shown here. Also, it is used that Jn−1 admitsagain a decomposition Jn−1 = Mn−1 · In−1 with Mn−1 a principal monomial idealsupported on some normal crossings divisor Dn−1 on V , and In−1 an ideal, theresidual factor of Jn−1.

There thus exists a sequence of blowups of V along regular centers transversalto En−1 and Dn−1 and their total transforms, contained in the locus top(Jn−1, c)of points where Jn−1 and its controlled transforms with respect to c have order ≥ c,and satisfying the requirements of a strong resolution so that the order of the weaktransform of In−1 drops eventually at all points below the maximum of the orderof In−1 at points a of V .

All this is well defined on the open neighbourhood U of a in W , but dependsa priori on the choice of the hypersurface V , since the centers are chosen locallyin each V . It is not clear that the local choices patch to give a globally definedcenter. One can show that this is indeed the case, even though the local ideals Jn−1

do depend on V . The argument relies on the fact that the centers of blowup areconstructed as the maximum locus of an invariant associated to Jn−1. By inductionon the ambient dimension, such an invariant exists for each Jn−1: In dimension 1,it is just the order. In higher dimension, it is a vector of orders of suitably definedcoefficient ideals. One then shows that the invariant of Jn−1 is independent of thechoice of V and defines an upper semicontinuous function on the stratum S. Itsmaximum locus is therefore well defined and closed. Again by induction on theambient dimension, it can be assumed that it is also regular and transversal to thedivisors En−1 and Dn−1.

By the assertion of the theorem in dimension n − 1, the order of the weaktransform of Jn−1 can be made everywhere smaller than c = o! by a suitablesequence of blowups. The sequence of blowups also transforms the original idealJ = M · I, producing controlled transforms of J and weak transforms of I. Theorder of I cannot increase in this sequence, if the centers are always chosen inside S.To achieve this inclusion a technical adjustment of the definition of coefficient idealsis required which will be omitted here. If the order of I drops, induction applies.So one is left to consider points where possible the order of the weak transformhas remained constant. There, one will use the commutativity of blowups with thepassage to coefficient ideals, Prop. 10.6: The coefficient ideal of the final transformof I will equal the controlled transform of the coefficient ideal Jn−1 of I. But theorder of this controlled transform has dropped below c, hence, by Prop. 10.9, alsothe order of the weak transform of I must have dropped. This proves the existenceof a resolution. �

Example 11.4. In the situation of the theorem, take W = A2, E = ∅, andJ = (x2y3) = 1 · I with control c+ = 1. Even though being a monomial, the ideal Jis not resolved yet, since it is not supported on the exceptional divisor. The idealJ has order 5 at 0, order 3 along the x-axis, and order 2 along the y-axis. Blow upA2 in 0. The controlled transform in the x-chart is J ! = (x4y3) with exceptionalfactor IE = (x4) and residual factor I1 = (y3), the strict transform of J . Thecontrolled transform in the y-chart is J ! = (x2y4) with exceptional factor IE = (y4)and residual factor I1 = (x2), the strict transform of J . One additional blowupresolves J .


Example 11.5. In the situation of the preceding example, replace E = ∅ byE = V (x+ y), respectively E = V (x+ y2), and resolve the ideal J .

Example 11.6. Take W = A3, E = V (x + z2), and J = (x2y3) with controlc+ = 1. The variety X defined by J is the union of the xz- and the yz-plane. Thetop locus of J is the z-axis, which is tangent to E. Therefore it is not allowed totake it as the center of the first blowup. The only possible center is the origin.Applying this blowup, the top locus of the controlled transform of J and the totaltransform of E have normal crossings, so that the top locus can now be chosen asthe center of the next blowup. Resolve the singularities of X.

Remark 11.7. The non-transversality of the candidate center of blowup withalready existing exceptional components is known as the transversality problem.The preceding example gives a first instance of the problem, see [EH02, Hau03] formore details.

Example 11.8. Take W = A3, E = D = ∅ and J = (x2 +yz). In characteristicdifferent from 2, the origin of A3 is the unique isolated singular point of the coneX defined by J . The ideal J has order 2 at 0. After blowing up the origin, thesingularity is resolved and the strict transform of J defines a regular surface. It istransversal to the exceptional divisor.

Example 11.9. Take W = A3, E = D = ∅, J = (x2 + yazb) with a, b ∈ N.According to the values of a and b, the top locus of J is either the origin, the y-or z-axis, or the union of the y- and the z-axis. If a + b ≥ 2, the yz-plane is ahypersurface of maximal contact for J at 0. The coefficient ideal is J1 = (yazb)(up to raising it to the square). This is a monomial ideal, but not supported yeton exceptional components. Resolve J .

Example 11.10. Take W = A4, E = D = V (yz), J = (x2 + y2z2(y + w)).The order of J at 0 is 2. The yzw-hyperplane has maximal contact with J at0, with coefficient ideal J1 = y2z2(y + w), in which the factor M1 = (y2z2) isexceptional and I1 = (y + w) is residual. Resolve J . The top locus of I1 is theplane V (y + w) ⊂ A3, hence not contained in the top locus of J . This technicalcomplication is handled by introducing an intermediate ideal, the companion ideal[Vil07, EH02].

Example 11.11. Compute the first few steps of the resolution process for thethree surfaces x2 + y2z, x2 + y3 + z5 and x3 + y4z5 + z11.

Example 11.12. Prove with all details the embedded resolution of plane curvesin characteristic zero according to the above description.

Example 11.13. Resolve the following items, taking into account differentcharacteristics.

(a) W = A2, E = ∅, J = I = (x2y3), c+ = 1.(b) W = A2, E = V (x), J = (x2y3), c+ = 1.(c) W = A2, E = V (xy), J = (x2y3), c+ = 1.(d) W = A3, E = V (x+ z2), J = (x2y3), c+ = 1.(e) W = A3, E = V (y + z), J = (x3 + (y + z)z2), c+ = 3.(f) W = A3, E = ∅, J = I = (x2 + yz), c+ = 2.(g) W = A3, E = ∅, J = I = (x3 + y2z2), c+ = 3.(h) W = A3, E = ∅, J = I = (x2 + xy2 + y5), c+ = 2.

52 HERWIG HAUSER

(i) W = A3, E = ∅, J = I = (x2 + xy3 + y5), c+ = 2.(j) W = A3, E = V (x+ y2), J = (x3 + (x+ y2)y4z5), c+ = 3.(k) W = A3, E = ∅, J = I = (x2 + y2z2(y + z)), c+ = 12.(l) W = A3, E = V (yz), J = (x2 + y2z2(y + z)), c+ = 2.(m) W = A4, E = V (yz), J = (x2 + y2z2(y + w)), c+ = 2.

Example 11.14. Consider J = I = (x2 + y7 + yz4) in A3, with E = ∅ andc+ = 2. Consider the sequence of three point blowups with the following centers.First the origin of A3, then the origin of the y-chart, then the origin of the z-chart.On the last blowup, consider the midpoint between the origin of the y- and thez-chart. Show that J is resolved at this point if the characteristic is zero or > 2.

Example 11.15. What happens in the preceding example if the characteristicis equal to 2? Describe in detail the behaviour of the first two components of theresolution invariant under the three local blowups.

Example 11.16. A combinatorial version of resolution is known as Hironaka’spolyhedral game: Let N be an integral convex polyhedron in Rn+, i.e., the positiveconvex hull N = conv(S+ Rn+) of a finite set S of points in Nn. Player A chooses asubset J of {1, . . . , n}, then player B chooses an element j ∈ J . After these moves,S is replaced by the set S′ of points α defined by α′i = αi if i 6= j and α′j =

∑k∈J αk,

giving rise to a new polyhedron N ′. Player A has won if after finitely many roundsthe polyhedron has become an orthant α+ Rn+. Player B can never win, but onlyprevent player A from winning. Show that player A has a winning strategy, firstwith and then without using induction on n [Spi83, Zei06].

12. Lecture XII: Positive Characteristic

The existence of the resolution of varieties of arbitrary dimension over a field ofpositive characteristic is still an open problem. For curves, there exist variousproofs [Kol07] chap. 1. For surfaces, the first proof of non-embedded resolutionwas given by Abhyankar in his thesis [Abh56]. Later, he proved embedded reso-lution, but the proof is scattered over several papers which sum up to over 500pages [Abh59, Abh64, Abh66b, Abh66a, Abh67, Abh98]. Cutkosky was able toshorten and simplify the argumens substantially [Cut11]. An invariant similar tothe one used by Abhyankar was developed independently by Zeillinger and Wagner[Zei05, Wag09, HW09]. Lipman gave an elegent proof of non-embedded resolutionfor arbitrary two-dimensional schemes [Lip78, Art86]. Hironaka proposed an invari-ant based on the Newton polyhedron which allows to prove embedded resolution ofsurfaces which are hypersurfaces [Hir84, Cos81, Hau00]. Hironaka’s invariant seemsto be restricted to work only for surfaces. The case of higher codimensional surfaceswas settled by Cossart-Jannsen-Saito, extending Hironaka’s invariant [CJS09]. Dif-ferent proofs were recently proposed by Benito-Villamayor and Kawanoue-Matsuki[BO12, KM12].

For three-folds, Abhyankar and Cossart gave partial results. Quite recently,Cossart-Piltant proved non-embedded resolution by a long case-by-case study [CP08,CP09]. See also [Cut09].

Programs and techniques for resolution in arbitrary dimension and charac-teristic have been developed quite recently, among others, by Hironaka, Teissier,Kuhlmann, Kawanoue-Matsuki, Benito-Bravo-Villamayor, Hauser-Schicho, Cossart[Hir12, Tei03, Kuh00, Kaw13, KM10, BO12, BV10, BV11, Hau10b, HS12, Cos11].


Remark 12.1. Let K be an algebraically closed field of prime characteristic p >0, and let X be an affine variety defined in An. The main problems already appearin the hypersurface case. Let f = 0 be an equation for X in An. Various propertiesof singularities used in the characteristic zero proof fail in positive characteristic:

(a) The top locus of f of points of maximal order need not be contained locallyin a regular hypersurface. Take the variety in A4 defined by f = x2+yz3+zw3+y7wover a field of characteristic 2 [Nar83, Mul83, Kaw13, Hau98].

(b) There exist sequences of blowups for which the sequence of points above agiven pointa where the order of the strict transforms of f has remained constant arenot contained eventually in the strict transforms of any regular local hypersurfacepassing through a.

(c) Derivatives cannot be used to construct hypersurfaces of maximal contact.(d) The characteristic zero invariant is no longer upper semicontinuous when

translated to positive characteristic.

Example 12.2. A typical situation where the characteristic zero resolutioninvariant does not work in positive characteristic is as follows. Take the polynomialf = x2 +y7 +yz4 over a field of characteristic 2. There exists a sequence of blowupsalong which the order of the strict transforms of f remains constant equal to 2 butwhere eventually the order of the residual factor of the coefficient ideal of f withrespect to a hypersurface of weak maximal contact increases.

The hypersurface V of W = A3 defined by x = 0 produces – up to raising theideal to the square – as coefficient ideal the ideal on A2 generated by y7 + yz4. Itsorder at the origin is 5.

Blow up A3 at the origin. In the y-chart W ′ of the blowup, the total transformof f is given by

f∗ = y2(x2 + y5 + y3z4),

with f ′ = x2 + y3(y2 + z4) the strict transform of f . It has order 2 at the originof W ′. The generator of the coefficient ideal of f ′ in V ′ : x = 0 decomposes into amonomial factor y3 and a residual factor y2 + z4. The order of the residual factorat the origin of W ′ is 2. Blow up W ′ along the z-axis, and consider the y-chartW ′′, with strict transform

f ′′ = x2 + y(y2 + z4),

of f . The residual factor of the coefficient ideal in V ′′ : x = 0 equals again y2 + z4.Blow up W ′′ at the origin and consider the z-chart W ′′′, with strict transform

f ′′′ = x2 + yz(y2 + z2).

The origin of W ′′′ is the intersection point of the two exceptional components y = 0and z = 0. The residual factor of the coefficient ideal in V ′′′ : x = 0 equals y2 + z2,of order 2 at the origin of W ′′′.

Blow up the origin of W ′′′ and consider the affine chart W (iv) given by thecoordinate transformation

x 7→ xz, y 7→ yz + z, and z 7→ z.

The origin of this chart is the midpoint of the new exceptional component. Thestrict transform of f ′′′ equals

f (iv) = x2 + (y + 1)z2y2,

54 HERWIG HAUSER

which, after the coordinate change x 7→ x+ yz, becomes

f (iv) = x2 + y3z2 + y2z2.

The order of the strict transforms of f has remained constant equal to 2 along thesequence of local blowups. The order of the residual factor of the associated coef-ficient ideal has decreased from 5 to 2 in the first blowup, then remained constantuntil the last blowup, where it increased from 2 to 3.

Remark 12.3. The preceding example shows that the order of the residualfactor of the coefficient ideal of the defining ideal of a singularity with respect to alocal hypersurface of weak maximal contact is not suited for an induction argumentas in the case of zero characteristic.

Definition 12.4. A hypersurface singularity X at a point a of affine spaceW = AnK over a field K of characteristic p is called purely inseparable of order pe

at a if there exist local coordinates x1, . . . , xn on W at a such that a = 0 and suchthat the local equation f of X at a is of the form

f = xpe

n + F (x1, . . . , xn−1)

for some e ≥ 1 and a polynomial F ∈ K[x1, . . . , xn−1] of order ≥ pe at a.

Proposition 12.5. For a purely inseparable hypersurface singularity X at a,the polynomial F is unique up to multiplication by units in the local ring OW,a andthe addition of pe-th powers in K[x1, . . . , xn−1].

Proof. Multiplication by units does not change the local geometry of X ata. A coordinate change in xn of the form xn 7→ xn + a(x1, . . . , xn−1) with a ∈K[x1, . . . , xn−1] transforms f into f = xp

e

n + a(x1, . . . , xn−1)pe

+ F (x1, . . . , xn−1).This implies the assertion. �

Definition 12.6. Let affine space W = An be equipped with an exceptionalnormal crossings divisor E produced by earlier blowups with multiplicities r1, . . . , rn.Let x1, . . . , xn be local coordinates at a point a of W such that E is defined at aby xr11 · · ·xrn

n = 0. Let f = xpe

n + F (x1, . . . , xn−1) define a purely inseparablesingularity X of order pe at the origin a = 0 of An such that F factorizes into

F (x1, . . . , xn−1) = xr11 · · ·xrn−1n−1 ·G(x1, . . . , xn−1).

The residual order of X at a with respect to E is the maximum of the orders of thepolynomials G at a over all choices of local coordinates such that f has the aboveform [Hir12, Hau10b].

Remark 12.7. The residual order can be defined for arbitrary singularities[Hau10b]. In characteristic zero, the definition coincides with the second componentof the local resolution invariant, defined by the choice of an osculating hypersurfaceor, more generally, of a hypersurface of weak maximal contact and the factorizationof the associated coefficient ideal.

Remark 12.8. In view of the preceding example, one is led to investigate thebehaviour of the residual order under blowup at points where the order of the singu-larity remains constant. Moh showed that it can increase at most by pe−1 [Moh87].Abhyankar seems to have observed already this bound in the case of surfaces. Hedefines a correction term ε taking values equal to 0 or pe−1 which is added to theresidual order according to the situation in order to make up for the occasional


increases of the residual order [Abh67, Cut11]. A similar construction has beenproposed by Zeillinger and Hauser-Wagner [Zei05, Wag09, HW09]. This allows, atleast for surfaces, to define a secondary invariant after the order of the singularity,the modified residual order of the coefficient ideal, which does not increase underblowup. The problem then is to handle the case where the order of the singularityand the modified residual order remain constant. It is not clear how to define athird invariant which manifests the improvement of the singularity.

Remark 12.9. Following ideas of Giraud, Cossart has studied the behaviourof the order of the Jacobian ideal of f , defined by certain partial derivatives of f .Again, it seems that hypersurfaces of maximal contact do not exist for this invariant[Gir75, Cos11]. There appeared promising recent approaches by Hironaka, usingthe machinery of differential operators in positive characteristic, by Villamayor andcollaborators using instead of the restriction to hypersurfaces of maximal contactprojections to regular hypersurfaces via elimination algebras, and by Kawanoue-Matsuki using their theory of idealistic filtrations and differential closures. Noneof these proposals has been able to produce an invariant or a resolution strategywhich works in positive characteristic for all dimensions.

Remark 12.10. Another approach consists in analyzing the singularities andblowups for which the residual order increases under blowup. This leads to thenotion of kangaroo singularities:

Definition 12.11. A hypersurface singularity X defined at a point a of affinespace W = AnK over a field K of characteristic p by a polynomial equation f = 0is called a kangaroo singularity if there exists a local blowup π : (W , a′) → (W,a)of W along a regular center Z contained in the top locus of X and transversal toan already existing exceptional normal crossings divisor E such that the order ofthe strict transform of X remains constant at a′ but the residual order of the stricttransform of f increases at a′. The point a′ is then called kangaroo point of Xabove a.

Remark 12.12. Kangaroo singularities can be defined for arbitrary singulari-ties. They have been characterized in all dimensions by Hauser [Hau10a, Hau10b].However, the knowledge of the algebraic structure of these singularities did not yetgive any hint how to overcome the obstruction caused by the increase of the residualorder.

Proposition 12.13. If a polynomial f = xpe

n + F (x1, . . . , xn−1) = xpe

n +xr11 · · ·x

rn−1n−1 ·G(x1, . . . , xn−1) defines a kangaroo singularity of order pe at 0 the sum

of the ri and of the order of G at 0 is divisible by pe, the sum of the residues of theexceptional multiplicities ri modulo pe is bounded by m · pe with m the number ofexceptional multiplicities not divisible by pe, and the initial form of F equals a spe-cific homogeneous polynomial prescribed by the situation. Any kangaroo point a′

of X above a lies outside the strict transform of the components of the exceptionaldivisor at a whose multiplicities are not a multiple of pe.

Remark 12.14. A more detailed description of kangaroo singularities and afurther discussion of typical characteristic p phenomena can be found in [Hau10a,Hau10b].

Example 12.15. Prove the resolution of plane curves in arbitary characteristicby using the order and the residual order as the resolution invariants.

56 HERWIG HAUSER

Example 12.16. Let K be an algebraically closed field of characteristic p >0. Develop a significant notion of resolution for elements of the quotient of ringsK[x, y]/K[xp, yp]. Then prove that such a resolution always exists.

Example 12.17. Consider the polynomial f = x2 + yz3 + zw3 + y7w on A4

over a ground field of characteristic 2. Its maximal order is 2, and the respectivetop locus is the image of the monomial curve (t32, t7, t19, t15), t ∈ K. The imagecurve has embedding dimension 4 at 0 and cannot be embedded locally at 0 intoa regular hypersurface of A4. Hence there is no hypersurface of maximal contactwith f locally at the origin.

Example 12.18. Find a surface X in positive characteristic and a sequence ofpoint blowups starting at a ∈ X so that some of the points above a where the orderof the weak transforms of X remains constant eventually leave the transforms ofany local regular hypersurface passing through a. Hint: You may use 72. or cookup your own example.

Example 12.19. Show that f = x2 + yz3 + zw3 + y7w has in characteristic2 top locus top(f) equal to the parametrized curve (t32, t7, t19, t15) in A4 [Nar83,Mul83, Kaw13].

Example 12.20. Show that f is not contained in the square of the ideal definingthe parametrized curve (t32, t7, t19, t15).

Example 12.21. Find the defining ideal for the image of the monomial curve(t32, t7, t19, t15) in A4. What is the local embedding dimension at 0?

Example 12.22. Show that f = x2 + yz3 + zw3 + y7w admits in character-istic 2 at the point 0 no local regular hypersurface of permanent maximal contact(i.e., whose successive strict transforms contain all points where the order of f hasremained constant in any sequence of blowups with regular centers inside the toplocus).

Example 12.23. Consider f = x2 + y7 + yz4 in characteristic 2. Show thatthere exists a sequence of point blowups for which f admits at the point 0 nolocal regular hypersurface whose transforms have weak maximal contact with thetransforms of f as long as the order of f remains equal to 2.

Example 12.24. Define the p-th order derivative of polynomials in K[x1, . . . , xn]for K a field of characteristic p.

Example 12.25. Construct a surface of order p5 in A3 for which the residualorder increases under blowup.

Example 12.26. Show that in f = xp + ypz with E = ∅ the residual orderalong the (closed) points of the z-axis is not equal to its value at the generic point.

Example 12.27. Let y1, . . . , ym be fixed coordinates, and consider a homoge-neous polynomial G(y) = yr · g(y) with r ∈ Nm and g(y) homogeneous of degreek. Let G+(y) be the polynomial obtained from G by the linear coordinate changeyi → yi + ym for i = 1, . . . ,m− 1. Show that the order of G+ along the ym-axis isat most k.

Example 12.28. Express the assertion of the preceding example through theinvertibility of a matrix of multinomial coefficients.


Example 12.29. Consider G(y, z) = yrzs∑ki=0

(k+ri+r

)yi(tz − y)k−i. Compute

for t ∈ K∗ the polynomial G+(y, z) = G(y + tz, z) and its order with respect to ymodulo p-th power polynomials.

Example 12.30. * Determine all homogeneous polynomialsG(y, z) = yrzsg(y, z)so that G+(y, z) has order k+ 1 with respect to y modulo p-th power polynomials.

Example 12.31. * Find a new sytematic proof for the embedded resolution ofsurfaces in three-space.

Example 12.32. Let G(x) be a polynomial in one variable over a field Kof characteristic p, of degree d and order k at 0. Let t ∈ K, and consider theequivalence class G of K(x+ t) in K[x]/K[xp] (i.e., consider K(x+ t) modulo p-thpower polynomials). What is the maximal order of G at 0? Describe all exampleswhere this maximum is achieved.

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